Introduction to the Bernoulli function
Peter Luschny

A companion to arXiv:2009.06743v2 [math.HO]

Numbering of formulas as in version 2.

Browsable at BernoulliFunctionNotebook.

Source at BernoulliFunctionSource.

Needs Wolfram Language kernel for Jupyter. See instructions at GitHub.

Preliminaries

(* This is used only for some fancy plot legends. *)
(* You will have to do this only once: *)
(* ResourceFunction["MaTeXInstall"][] *)
(* Thereafter you have to execute only: *)
<<MaTeX`
(* MaTeX["x^2"] *)
(* But you not really need these fancy plot legends. *)
(* Comment them out to use this notebbok without MaTex. *)
MaTeX@HoldForm[Integrate[Sin[x], {x, 0, Infinity}]]
(* http://szhorvat.net/pelican/latex-typesetting-in-mathematica.html *)

(* Always execute first. *)

(* 0^0 = 1 *)
Unprotect[Power]; Power[0, 0] = 1; 

(* tau = 2*Pi*I *)
tau := 2*Pi*I;
\[Tau][s_] := tau^(-s) + (-tau)^(-s); 

(* Macro to get a better format *)
Poly[a_] := PolynomialForm[Collect[Expand[Simplify[FunctionExpand[a]]], x], 
            TraditionalOrder -> True];

(* Macro to display a sequence table *)
SeqGrid[T_] := With[{y = Length[T]},
               Grid[{Table[i, {i, 0, y-1}], T}, Frame -> All]];
SeqGrid1[T_] := With[{y = Length[T]},
               Grid[{Table[i, {i, y}], T}, Frame -> All]];             
SeqGrid2[T_] := With[{y = Length[T] - 1},
               Grid[{Table[2 i, {i, 0, y}], T}, Frame -> All]];

Stieljes constants and zeta function

Formula 1

Laurent series of the Riemann zeta function

Series[Zeta[s] - 1/(s - 1), {s, 1, 5}] // TeXForm

$$\gamma -(s-1) \gamma _1+\frac{1}{2} (s-1)^2 \gamma _2-\frac{1}{6} (s-1)^3 \gamma _3+\frac{1}{24} (s-1)^4 \gamma _4-\frac{1}{120} (s-1)^5 \gamma _5+O\left((s-1)^6\right)$$

Formula 2

Stieltjes constants, defined via integral

gamma[s0_, prec_] :=  (* parameter 'prec' for setting the precision *)
Module[{s, z}, {s} = SetPrecision[{s0}, prec $MachinePrecision];
    (-4 Pi /(s + 1)) NIntegrate[Log[1/2 + I z]^(s + 1)/(Exp[-Pi z] + Exp[Pi z])^2,
    {z, 0, Infinity},  WorkingPrecision -> prec $MachinePrecision ]];

Table[gamma[n, 2], {n, 0, 5}]

Table[Re[gamma[n, 2]], {n, 0, 5}]  // MatrixForm  
Table[StieltjesGamma[k], {k, 0, 5}] // N // SeqGrid

Bernoulli constants and Bernoulli function

Formula 3

Bernoulli constants, defined via integral

HoldForm[2 Pi Integrate[Log[1/2 + I z]^s / (Exp[-Pi z] + Exp[Pi z])^2, 
     {z, -Infinity, Infinity}]] // TeXForm

$$2 \pi \int_{-\infty }^{\infty } \frac{\log ^s\left(\frac{1}{2}+i z\right)}{(\exp (-\pi z)+\exp (\pi z))^2} \, dz$$

beta[s0_, prec_] :=  (* parameter 'prec' for setting the precision *)
Module[{s, z}, {s} = SetPrecision[{s0}, prec $MachinePrecision];
    4 Pi NIntegrate[Log[1/2 + I z]^s / (Exp[-Pi z] + Exp[Pi z])^2,
    {z, 0, Infinity},  WorkingPrecision -> prec $MachinePrecision ]];

Table[beta[n, 2], {n, 0, 5}] // MatrixForm

Table[Re[beta[n, 2]], {n, 0, 5}] // MatrixForm // TeXForm

$$\left( \begin{array}{c} 1.0000000000000000000000000000000 \\ -0.57721566490153286060651209008240 \\ 0.14563169096735344972117275174980 \\ 0.029071089578616955453591158105638 \\ -0.008215337681213383464640186171014 \\ -0.011626850327336500287340850887630 \\ \end{array} \right)$$

Memoization of the real part of the Bernoulli constants with single precision. (For quick checking and plotting, for higher precision use the two parameter form.)

beta[n_] := beta[n] = Re[beta[n, 1]];

Plot[beta[s], {s, -0.6, 4}, WorkingPrecision -> 30,
Epilog -> {PointSize[0.01], Red, Point[Table[{k, beta[k]}, {k, 0, 4}]]},  
PlotLegends -> Placed["Bernoulli constants", {Scaled[{0.9, 0.9}], {0.9, 0.9}}]]
(* Export["Fig2BernoulliConstants.eps", %] *)

Formula 4

Bernoulli function, definition.

Bbeta[s_] := Sum[beta[j] s^j / j!, {j, 0, 50}];

Table[Bbeta[n], {n, 0, 10, 2}] // N // SeqGrid2

Table[BernoulliB[n, 1], {n, 0, 10, 2}] // N // SeqGrid2

Formula 5

Bernoulli function using the Riemann zeta representation.

Limit[-n Zeta[1 - n], n -> 0]
B[s_] := -s Zeta[1 - s];
B[0] := 1;

Clear[s]; B[s] // FullSimplify // TeXForm

1

$$-s \zeta (1-s)$$

From now onwards we use Mathematica's efficient implementation of the Zeta function for the Bernoulli function.

Plot[B[s], {s, 0, 12}, PlotRange -> {-1/3, 1},
Epilog -> {PointSize[0.01], Red, Point[Table[{k,BernoulliB[k,1]}, {k, 0, 12}]]}]

Bernoulli function on the critical line and the zeta zeros which are the same as the Bernoulli zeros.

ReImPlot[B[1/2 + I t], {t, 20, 52},
Epilog -> {PointSize[0.01], Red, Point[Table[{Im[ZetaZero[k]], 0}, {k, 11}]]}]
(* Export["Fig10BernoullisZetaZeros.eps", %] *)

Table[Im[ZetaZero[n]], {n, 1, 11}] // N // SeqGrid1

Bernoulli function rises up at Riemann's critical line.

ComplexPlot3D[B[s], {s, -60 - 40 I, 1/2 + 40 I}, 
WorkingPrecision -> 40,
AxesLabel -> {"Re(s)", "Im(s)"},
ImageSize -> Large,
ViewPoint -> {2.6, -0.9, 0.6} ]
(* Export["Fig27BernoulliTsunami.pdf", %] *)

ComplexStreamPlot[B[s], {s, 0 - 8 I, 16 + 8 I}]
(* Export["Fig11BernoulliPhasePortrait.eps", %] *)

ComplexPlot3D [B[s], {s, 0 - 8 I, 16 + 8 I}, 
ColorFunction -> "CyclicArg", ViewProjection -> "Perspective"]

ComplexPlot[B[s], {s, 0 - 8 I, 16 + 8 I}, ColorFunction -> "CyclicLogAbs"]

Formula 6, 7

Table[B[n], {n, 0, 12}] // SeqGrid

Formula 8

Series[-s Zeta[1 - s], {s, 0, 4}] // TeXForm
% // Normal // N // Simplify // TeXForm

$$1-\gamma s-s^2 \gamma _1-\frac{s^3 \gamma _2}{2}-\frac{s^4 \gamma _3}{6}+O\left(s^5\right)$$

$$\text{-0.000342306 s${}^{\wedge}$4+0.00484518 s${}^{\wedge}$3+0.0728158 s${}^{\wedge}$2-0.577216 s+1.}$$

Formula 9

Bernoulli function Taylor expansion based on the Bernoulli constants.

Series[N[Bbeta[z]], {z, 0, 4}] // TeXForm

$$1.-0.577216 z+0.0728158 z^2+0.00484518 z^3-0.000342306 z^4+O\left(z^5\right)$$

Formula 10

Alternative formula for the Bernoulli constants.

beta2[s0_, prec_] :=  (* parameter 'prec' for setting the precision *)
Module[{s, z}, {s} = SetPrecision[{s0}, prec $MachinePrecision];
    Pi NIntegrate[Re[Log[1/2 + I z]^s Sech[Pi z]^2],
    {z, 0, Infinity},  WorkingPrecision -> prec $MachinePrecision ]];

Table[beta2[n, 2], {n, 0, 5}] // MatrixForm // TeXForm

$$\left( \begin{array}{c} 1.0000000000000000000000000000000 \\ -0.57721566490153286060651209008240 \\ 0.14563169096735344972117275174980 \\ 0.029071089578616955453591158105638 \\ -0.008215337681213383464640186171014 \\ -0.011626850327336500287340850887630 \\ \end{array} \right)$$

Formula 11

sigma[n_, z_] := Re[Log[1/2 + I z]^n];
ComplexExpand[sigma[n, z]] // FullSimplify // TeXForm

$$\left(\frac{1}{4} \log ^2\left(z^2+\frac{1}{4}\right)+\arg \left(\frac{1}{2}+i z\right)^2\right)^{n/2} \cos \left(n \arg \left(\log \left(\frac{1}{2}+i z\right)\right)\right)$$

Formula 12

a[z_] := Log[z^2 + 1/4] / 2;
b[z_] := ArcTan[2 z];

sigmasum[n_, z_] := Sum[(-1)^k Binomial[n, 2 k] a[z]^(n - 2 k) b[z]^(2 k),
                    {k, 0, Floor[n / 2]}]
                    
Table[sigmasum[n, z], {n, 0, 4}] // TableForm // TeXForm

$$\begin{array}{c} 1 \\ \frac{1}{2} \log \left(z^2+\frac{1}{4}\right) \\ \frac{1}{4} \log ^2\left(z^2+\frac{1}{4}\right)-\tan ^{-1}(2 z)^2 \\ \frac{1}{8} \log ^3\left(z^2+\frac{1}{4}\right)-\frac{3}{2} \log \left(z^2+\frac{1}{4}\right) \tan ^{-1}(2 z)^2 \\ \frac{1}{16} \log ^4\left(z^2+\frac{1}{4}\right)-\frac{3}{2} \log ^2\left(z^2+\frac{1}{4}\right) \tan ^{-1}(2 z)^2+\tan ^{-1}(2 z)^4 \\ \end{array}$$

Formula 13

beta2[n_] := Pi NIntegrate[sigma[n, z] / Cosh[Pi z]^2, {z, 0, 10}];
beta3[n_] := Pi NIntegrate[sigmasum[n, z] / Cosh[Pi z]^2, {z, 0, 10}];

Table[beta[n],  {n, 0, 5}] // N // Chop
Table[beta2[n], {n, 0, 5}] // N // Chop
Table[beta3[n], {n, 0, 5}] // N // Chop

{1., -0.577216, 0.145632, 0.0290711, -0.00821534, -0.0116269}

{1., -0.577216, 0.145632, 0.0290711, -0.00821534, -0.0116269}

{1., -0.577216, 0.145632, 0.0290711, -0.00821534, -0.0116269}

Formula 14

gamma2[n_] := -(Pi / (n + 1)) NIntegrate[Re[sigma[n + 1, z] / Cosh[Pi z]^2], 
                              {z, 0, 10}];
              
Table[gamma2[n],  {n, 0, 5}] // N 
Table[StieltjesGamma[n], {n, 0, 5}] // N

{0.577216, -0.0728158, -0.00969036, 0.00205383, 0.00232537, 0.000793324}

{0.577216, -0.0728158, -0.00969036, 0.00205383, 0.00232537, 0.000793324}

Some special integral formulas

Formula 15

Euler's gamma.

eulergamma = -(Pi/2) NIntegrate[Log[z^2 + 1/4] / Cosh[Pi z]^2, {z, 0, Infinity}]

0.577216

Formulas 16, 17, 18, 19, 20

Some Bernoulli constants.

With[{a = Log[z^2 + 1/4] / 2, b = ArcTan[2 z], c = Cosh[Pi z]}, {
Pi * NIntegrate[a / c^2, {z, 0, Infinity}],
Pi * NIntegrate[(a^2 - b^2) / c^2, {z, 0, Infinity}],
Pi * NIntegrate[(a^3 - 3 a b^2) / c^2, {z, 0, Infinity}],
Pi * NIntegrate[(a^4 - 6 a^2 b^2 + b^4) / c^2, {z, 0, Infinity}],
Pi * NIntegrate[(a^5 - 10 a^3 b^2 + 5 a b^4) / c^2, {z, 0, Infinity}]}]

{-0.577216, 0.145632, 0.0290711, -0.00821534, -0.0116269}

Generalized Bernoulli constants

Formula 21

Laurent expansion of the Hurwitz zeta function.

Series[HurwitzZeta[s, v] - 1/(s - 1), {s, 1, 4}] // TeXForm

$$-\psi ^{(0)}(v)-(s-1) \gamma _1(v)+\frac{1}{2} (s-1)^2 \gamma _2(v)-\frac{1}{6} (s-1)^3 \gamma _3(v)+\frac{1}{24} (s-1)^4 \gamma _4(v)+O\left((s-1)^5\right)$$

Formula 22

Generalized Stieltjes constants, integral representation.

IntGamma[n_, v_] := -(Pi / (2 (n + 1))) * 
     NIntegrate[Log[v - 1/2 + I z]^(n + 1) / Cosh[Pi z]^2, 
     {z, -Infinity, Infinity}, WorkingPrecision -> 30];

Table[{Chop[IntGamma[s, 1]], N[StieltjesGamma[s], 30]}, {s, 0, 4}] // TableForm

Formula 23

Generalized Bernoulli constants, definition via integral.

IntBeta[s_, v_] := 2 Pi *
        NIntegrate[Log[v - 1/2 + I z]^s / (Exp[Pi z] + Exp[-Pi z])^2, 
        {z, -Infinity, Infinity}, WorkingPrecision -> 30];

Table[{Chop[IntBeta[s+1, 1]], N[-(s + 1) StieltjesGamma[s], 30]}, {s, 0, 4}] // TableForm

Generalized Bernoulli constants, definition via Stieltjes constants.

BernBeta[s_, v_] := -s StieltjesGamma[s - 1, v];
BernBeta[0, _] := 1;

Table[BernBeta[s, 1], {s, 0, 4}] // N

{1., -0.577216, 0.145632, 0.0290711, -0.00821534}

IntBeta[Pi, 1]

-0.0288694485250208576507944887079

Mathematica can't do this!

(* BernBeta[Pi, 1] *)

The generalized Bernoulli function

Formula 24, 25, 26

Generalized Bernoulli function. (Closing definition gap with limiting value.)

Limit[-s HurwitzZeta[1 - s, v], s -> 0]
 
Bg[s_, v_] := If[s == 0, 1, -s*HurwitzZeta[1 - s, v]]; 
Clear[s, v]; Bg[s, v] // FullSimplify // TeXForm

1

$$\text{If}[s=0,1,-s \zeta (1-s,v)]$$

atext := MaTeX["B(s, 1/4)"];
btext := MaTeX["B(s, 1/2)"];
ctext := MaTeX["B(s, 3/4)"];

Plot[{Bg[s, 1/4], Bg[s, 1/2], Bg[s, 3/4]}, {s, 1, 9}, Filling -> {1 -> {2}},
PlotTheme -> {"Thin"}, WorkingPrecision -> 40, ImageSize -> Large,
PlotLegends -> Placed[{atext, btext, ctext}, Above],
PlotLabel -> "Generalized Bernoulli function" ]
(* Export["FigXXGeneralizedBernoulliFunction.eps", %] *)

Formula 27

Bernoulli polynomials, explicit formula

Bpoly[n_, x_] := Sum[(-1)^j Binomial[n, j] B[j] x^(n-j), {j, 0, n}];
Clear[n, x]; Bpoly[n, x] // FullSimplify // TeXForm

$$\sum _{j=0}^n (-1)^{j+1} j \zeta (1-j) \binom{n}{j} x^{n-j}$$

{Table[Expand[Bpoly[n, x]], {n, 0, 5}] // MatrixForm , 
 Table[BernoulliB[n, x],    {n, 0, 5}] // MatrixForm }

V := Table[FunctionExpand[Bg[n, x]], {n, 1, 6}];
W := Table[BernoulliB[n, x], {n, 1, 6}]; 
Grid[{V, W}, Frame -> All]

Plot[V, {x, -1/2, 3/2}, PlotRange -> {-0.2, 0.2}]

Integral formulas for the Bernoulli function

Formula 28, 29

The Jensen integral of the Bernoulli function.

HoldForm[2 Pi Integrate[(1/2 +  I z)^s / (Exp[-Pi z] + Exp[Pi z])^2, 
     {z, -Infinity, Infinity}]] // TeXForm

$$2 \pi \int_{-\infty }^{\infty } \frac{\left(\frac{1}{2}+i z\right)^s}{(\exp (-\pi z)+\exp (\pi z))^2} \, dz$$

Making the integrand real.

f[v_, s_, z_] := (v - 1/2 +  I z)^s / (Exp[-Pi z] + Exp[Pi z])^2;
Clear[v, s, z]; f[v, s, z]     // FullSimplify // TeXForm
ComplexExpand[f[v, s, z]]      // FullSimplify // TeXForm
ComplexExpand[Re[f[v, s, z]]]  // FullSimplify // TeXForm

$$\frac{1}{4} \text{sech}^2(\pi z) \left(v+i z-\frac{1}{2}\right)^s$$

$$\frac{\left(\left(v-\frac{1}{2}\right)^2+z^2\right)^{s/2} e^{2 \pi z+i s \arg \left(v+i z-\frac{1}{2}\right)}}{\left(e^{2 \pi z}+1\right)^2}$$

$$\frac{1}{4} \text{sech}^2(\pi z) \left((v-1) v+z^2+\frac{1}{4}\right)^{s/2} \cos \left(s \arg \left(v+i z-\frac{1}{2}\right)\right)$$

The real Bernoulli function, via Jensen integral.

JFR[s0_, z0_, prec_] := Module[{s, z}, 
     {s, z} = SetPrecision[{s0, z0}, prec $MachinePrecision];
     Block[{$MinPrecision = prec $MachinePrecision, 
     $MaxPrecision = prec $MachinePrecision},
     (z^2 + 1/4)^(s/2) Cos[s Arg[I z + 1/2]] Sech[Pi z]^2 ]];

JRIntegral[s0_, prec_] := Module[{s}, 
     {s} = SetPrecision[{s0}, prec $MachinePrecision];
     Pi NIntegrate[JFR[s, z, prec], {z, 0, Infinity}, 
     WorkingPrecision -> prec $MachinePrecision ]];

JRIntegral[2.5, 4]
Precision[%]

0.06371300472458258987385746727676186367256246150243249208661374587

63.8184

An arbitrary precision version of the Bernoulli function via zeta function.

BN[s0_, prec_] := Module[{s},{s} = SetPrecision[{s0}, prec $MachinePrecision];
       Block[{$MinPrecision = prec $MachinePrecision, 
       $MaxPrecision = prec $MachinePrecision},
       -s Zeta[1 - s] ]];
       
BN[2.5, 4] 
Precision[%]

0.06371300472458258987385746727676186367256246150243249208661374587

63.8184

Test cases, even indexed Bernoulli numbers, Jensen integral versus zeta version

Table[{JRIntegral[n, 2], N[BernoulliB[n], 32]}, {n, 0, 8, 2}]

{{1.0000000000000000000000000000000, 1.0000000000000000000000000000000}, 
 
>   {0.16666666666666666666666666666667, 0.16666666666666666666666666666667}, 
 
>   {-0.033333333333333333333333333333333, -0.033333333333333333333333333333333}, 
 
>   {0.023809523809523809523809523809524, 0.023809523809523809523809523809524}, 
 
>   {-0.033333333333333333333333333333333, -0.033333333333333333333333333333333}}

Table[{JRIntegral[n + 1/2, 2], BN[n + 1/2, 2]}, {n, 1, 8, 2}]

{{0.31182933746603184902596008809557, 0.31182933746603184902596008809557}, 
 
>   {-0.029809250722476156898254984599206, -0.029809250722476156898254984599206}, 
 
>   {0.017004180859687086146533412098215, 0.017004180859687086146533412098215}, 
 
>   {-0.020600759546526515688172706141941, -0.020600759546526515688172706141941}}

The complex Bernoulli function, via Jensen integral.

Formula 30

The generalized complex Bernoulli function.

JFC[s0_, z0_, prec_] := Module[{s, z}, 
     {s, z} = SetPrecision[{s0, z0}, prec $MachinePrecision];
     Block[{$MinPrecision = prec $MachinePrecision, 
     $MaxPrecision = prec $MachinePrecision},
     (1/2 + I z)^s / (Exp[-Pi z] + Exp[Pi z])^2 ]];

JCIntegral[s0_, prec_] := Module[{s}, 
     {s} = SetPrecision[{s0}, prec $MachinePrecision];
     4 Pi NIntegrate[JFC[s, z, prec], {z, 0, Infinity}, 
     WorkingPrecision -> prec $MachinePrecision ]];

JCIntegral[2.5, 2]
Precision[%]

Do you know the imaginary part of the Bernoulli numbers?

Table[JCIntegral[n, 2], {n, 0, 8, 2}] // MatrixForm

Can you identify more values?

Table[Im[JCIntegral[n, 2]], {n, 0, 8, 2}] // MatrixForm // TeXForm
{ 0, Log[2]/Pi, 1/42 } // N

$$\left( \begin{array}{c} 0 \\ 0.22063560015265159339645643211800 \\ 0.023089395969762999080708462841876 \\ 0.0038084668530144729535940040711306 \\ 0.00071731231136442466964684684256360 \\ \end{array} \right)$$

{0., 0.220636, 0.0238095}

ReImPlot[JCIntegral[x, 1], {x, -1, 8}, PlotRange -> {-0.4, 1}]

The Hurwitz-Bernoulli function

Formula 31

The Hurwitz–Bernoulli function.

PL[s_, v_] := (-s! / (2 Pi)^s) PolyLog[s, Exp[2 Pi I v]];
HB[s_, v_] := Exp[-I Pi s / 2] PL[s, v] + Exp[I Pi s / 2] PL[s, 1 - v];

Clear[s, v]; HB[s, v] // FullSimplify // TeXForm

$$-e^{-\frac{1}{2} i \pi s} (2 \pi )^{-s} s! \left(e^{i \pi s} \text{Li}_s\left(e^{-2 i \pi v}\right)+\text{Li}_s\left(e^{2 i \pi v}\right)\right)$$

Formula 32

The Hurwitz–Bernoulli function represents the Bernoulli function for 0 ≤ v ≤ 1 and s > 1.

Plot[{Bg[s, 3/4], HB[s, 3/4]}, {s, 1, 4}, 
PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.7}], {0.9, 0.7}}]]

The Hurwitz–Bernoulli functions with s = 2 + k/6 for 0 ≤ k ≤ 6 deform $B_2(x)$ into $B_3(x)$.

deform := Table[HB[2 + k/6, x], {k, 0, 6}];
Plot[deform, {x, 0, 1}]
(* Export["Fig4HurwitzBernoulliFunction.eps", %] *)

The central Bernoulli function

Formula 33

The case v = 1 in the Hurwitz–Bernoulli function.

B1[s_] := -2 s! PolyLog[s, 1] Cos[s Pi/2] / (2 Pi)^s;
{Limit[B1[n], n->0], Limit[B1[n], n->1], Table[B1[n], {n, 2, 12}]} // TeXForm

$$\left\{1,\frac{1}{2},\left\{\frac{1}{6},0,-\frac{1}{30},0,\frac{1}{42},0,-\frac{1}{30},0,\frac{5}{66},0,-\frac{691}{2730}\right\}\right\}$$

Formula 34

Representation of the Bernoulli function by the PolyLog.

Plot[{B[s], B1[s]}, {s, 1, 4}, 
PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.7}], {0.9, 0.7}}]]

Formula 35

The case v = 1/2 in the Hurwitz–Bernoulli function: representation of the central Bernoulli function by the PolyLog.

Bch[s_] := -2 s! PolyLog[s, -1] Cos[s Pi/2] / (2 Pi)^s;

Table[Bch[n], {n, 0, 12}] // TeXForm

$$\left\{1,0,-\frac{1}{12},0,\frac{7}{240},0,-\frac{31}{1344},0,\frac{127}{3840},0,-\frac{2555}{33792},0,\frac{1414477}{5591040}\right\}$$

Series[Bch[s], {s, 0, 6}] // Normal // N // TeXForm

$$0.00267343 s^6-0.0191478 s^5+0.111237 s^4-0.484432 s^3+1.35346 s^2-1.96351 s+1.$$

Formula 36

The central Bernoulli function, used as definition.

Bc[s_] := Bg[s, 1/2]; 
Clear[s]; Bc[s] // TeXForm

$$\text{If}\left[s=0,1,-s \zeta \left(1-s,\frac{1}{2}\right)\right]$$

Series[Bc[s], {s, 0, 6}] // Normal // N // TeXForm

$$0.00267343 s^6-0.0191478 s^5+0.111237 s^4-0.484432 s^3+1.35346 s^2-1.96351 s+1.$$

Table[Bc[n], {n, 0, 12, 2}] // TeXForm

$$\left\{1,-\frac{1}{12},\frac{7}{240},-\frac{31}{1344},\frac{127}{3840},-\frac{2555}{33792},\frac{1414477}{5591040}\right\}$$

Plot[{B[s], Bc[s]}, {s, 0, 11},  PlotRange -> {-0.1, 1}, WorkingPrecision -> 40,
PlotTheme -> {"Thin"}, Filling -> {1 -> {2}},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.7}], {0.9, 0.7}}]]
(* Export["Fig5CenteredBernoulliFunction.eps", %] *)

Formula 36a

The central Bernoulli function, via Bernoulli function.

BcB[s_] := B[s] (2^(1 - s) - 1);

Table[BcB[n], {n, 0, 12, 2}] // TeXForm

$$\left\{1,-\frac{1}{12},\frac{7}{240},-\frac{31}{1344},\frac{127}{3840},-\frac{2555}{33792},\frac{1414477}{5591040}\right\}$$

Series[BcB[s], {s, 0, 6}] // Normal // N // TeXForm

$$0.00267343 s^6-0.0191478 s^5+0.111237 s^4-0.484432 s^3+1.35346 s^2-1.96351 s+1.$$

Formula 37

Jensen integral of the central Bernoulli function for even integer n.

f[n_, z_] := ((I z)^n / (Exp[-Pi z] + Exp[Pi z]))^2;
Clear[n, z]; f[n, z] // FullSimplify // TeXForm

$$\frac{1}{4} (i z)^{2 n} \text{sech}^2(\pi z)$$

JInt[n_] := 2 Pi NIntegrate[f[n, z], {z, -Infinity, Infinity}, 
              WorkingPrecision -> 30] 
Table[JInt[n], {n, 0, 6}] // MatrixForm // TeXForm

$$\left( \begin{array}{c} 1.0000000000000000000000000000 \\ -0.0833333333333333333333333333333 \\ 0.0291666666666666666666666666667 \\ -0.0230654761904761904761904761905 \\ 0.0330729166666666666666666666667 \\ -0.0756096117424242424242424242424 \\ 0.252989962511446886446886446886 \\ \end{array} \right)$$

Computation of the even indexed Bernoulli numbers via the integral representation of the centered Bernoulli function.

NBInt[n_] := Pi ((-4)^(n + 1) / (4^n - 2)) NIntegrate[
            (z^n / (Exp[-Pi z] + Exp[Pi z]))^2, {z, 0, Infinity}, 
            Method -> {"GlobalAdaptive", Method -> "GaussKronrodRule"}, 
            MaxRecursion -> 100, WorkingPrecision -> 30]

{Table[NBInt[n], {n, 0, 6}] // MatrixForm,
Table[BernoulliB[2 n, 1], {n, 0, 6}] // N // MatrixForm } // TeXForm

$$\left\{\left( \begin{array}{c} 1.0000000000000000000000000000 \\ 0.166666666666666666666666666667 \\ -0.0333333333333333333333333333333 \\ 0.0238095238095238095238095238095 \\ -0.0333333333333333333333333333333 \\ 0.0757575757575757575757575757576 \\ -0.253113553113553113553113553114 \\ \end{array} \right),\left( \begin{array}{c} 1. \\ 0.166667 \\ -0.0333333 \\ 0.0238095 \\ -0.0333333 \\ 0.0757576 \\ -0.253114 \\ \end{array} \right)\right\}$$

n = 500;
N[BernoulliB[2 n], 30]
NBInt[n]

                                   1769
-5.31870446941552203648291374377 10

                                   1769
-5.31870446941552203648291374377 10

Formula 38

The secant decomposition of B(s) and the cosecant numbers A001896.

CosecInt[n_] := 2 Pi (-1)^n NIntegrate[
            (z^n / (Exp[-Pi z] + Exp[Pi z]))^2, {z, -Infinity, Infinity}, 
            Method -> {"GlobalAdaptive", Method -> "GaussKronrodRule"}, 
            MaxRecursion -> 100, WorkingPrecision -> 30]

Table[Rationalize[CosecInt[n]], {n, 0, 8}] // TeXForm

$$\left\{1,-\frac{1}{12},\frac{7}{240},-\frac{31}{1344},\frac{127}{3840},-\frac{2555}{33792},\frac{1414477}{5591040},-\frac{57337}{49152},\frac{118518239}{16711680}\right\}$$

BInt[s_] := Pi (Cos[Pi s / 2] / (2^(1 - s) - 1)) NIntegrate[ 
            z^s Sech[Pi z]^2, {z, 0, Infinity}, 
            Method -> {"GlobalAdaptive", Method -> "GaussKronrodRule"},
            WorkingPrecision -> 30, MaxRecursion -> 100]

Table[{Chop[N[B[n + 1/3], 25]], BInt[n + 1/3]}, {n, 2, 12}] // TeXForm

$$\left( \begin{array}{cc} 0.09347643656461653609586977 & 0.0934764365646165360958697658931 \\ -0.02321317157144160106800172 & -0.0232131715714416010680017186069 \\ -0.02567996314147641945039502 & -0.0256799631414764194503950235983 \\ 0.01218267560510201354599632 & 0.0121826756051020135459963211715 \\ 0.02095856813231439749162598 & 0.0209585681323143974916259834560 \\ -0.01402535703171405419441162 & -0.0140253570317140541944116210420 \\ -0.03211189995488664768966205 & -0.0321118999548866476896620534172 \\ 0.02749516420035317359639524 & 0.0274951642003531735963952380974 \\ 0.07825826130216604316107834 & 0.0782582613021660431610783363366 \\ -0.08146583848027172936296030 & -0.0814658384802717293629603015525 \\ -0.2769181636533657312698068 & -0.276918163653365731269806817235 \\ \end{array} \right)$$

atext := MaTeX["\\pi \\cos(\\pi s/2)/(2^{1 - s} - 1)"];
btext := MaTeX@HoldForm[Integrate[z^s Sech[Pi z]^2, {z, 0, Infinity}]]
atext
btext

Bosc[s_] := Pi (Cos[Pi s / 2] / (2^(1 - s) - 1)) ;
Plot[{Bosc[s], B[s] / Bosc[s]}, {s, -1, 17}, PlotRange -> {-4, 8},
PlotLegends -> Placed[{atext, btext}, Above]]
(* Export["Fig35SecantDecomposition.eps", %] *)

The central Bernoulli polynomials

Formula 39

Central Bernoulli numbers.

BcNum[n_] := 2^n Bg[n, 1/2];
Table[BcNum[n], {n, 0, 12}] // TeXForm

$$\left\{1,0,-\frac{1}{3},0,\frac{7}{15},0,-\frac{31}{21},0,\frac{127}{15},0,-\frac{2555}{33},0,\frac{1414477}{1365}\right\}$$

BcLi[n_] := 2*(-1)^(n+1)*Pi^(-2*n)*(2*n)!*PolyLog[2*n, -1];
Table[BcLi[n], {n, 0, 8}] // TeXForm

$$\left\{1,-\frac{1}{3},\frac{7}{15},-\frac{31}{21},\frac{127}{15},-\frac{2555}{33},\frac{1414477}{1365},-\frac{57337}{3},\frac{118518239}{255}\right\}$$

Clausen[0] = 1; 
Clausen[n_] := Times @@ (Select[Divisors[n+1], PrimeQ[# + 1] &] + 1);
Table[Clausen[n], {n, 0, 11}]

{1, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730}

BcL[n_] := (2*n)!*PolyLog[2*n, 1]*Pi^(-2*n)*Clausen[2*n - 1] / 2;
Table[BcL[n], {n, 1, 18}]

{1, 4, 16, 64, 1280, 707584, 28672, 59260928, 2874867712, 45773225984, 896021823488, 
 
>   991382852337664, 143497256501248, 1593799350268461056, 2312797281748024033280, 
 
>   8277820436597920759808, 11071085050982544965632, 452092922822895257024443973632}

Table[Bg[n, 2], {n, 1, 18}]

BcLi[n_] := n!*PolyLog[n, 1]*Pi^(-n);
Table[BcLi[n], {n, 1, 8}] // TeXForm

$$\left\{\infty ,\frac{1}{3},\frac{6 \zeta (3)}{\pi ^3},\frac{4}{15},\frac{120 \zeta (5)}{\pi ^5},\frac{16}{21},\frac{5040 \zeta (7)}{\pi ^7},\frac{64}{15}\right\}$$

Formula 40

Central Bernoulli polynomials.

Bcpoly[n_, x_] := Sum[Binomial[n, k] BcNum[k] x^(n - k), {k, 0, n}];
(* Clear[n, x]; Bc[n, x] // FullSimplify // TeXForm *)

Table[Bcpoly[n, x], {n, 0, 6}] // MatrixForm

Table[Bcpoly[n, -1], {n, 0, 12}] 
Table[Bcpoly[n, 1], {n, 0, 12}] 
Table[Bcpoly[n, 0], {n, 0, 12}]

NormBcpoly := Table[Expand[FullSimplify[Bcpoly[n, x] / n!]], {n, 0, 7}] 
Plot[NormBcpoly, {x, -2.5, 2.5}, PlotRange -> {-0.2, 0.2}]

Formula 41

Going halves.

s[n_] := Sum[Binomial[n, k] 2^k Bg[k, 1/2], {k, 0, n}];
Table[s[n], {n, 0, 12}] // TeXForm

$$\left\{1,1,\frac{2}{3},0,-\frac{8}{15},0,\frac{32}{21},0,-\frac{128}{15},0,\frac{2560}{33},0,-\frac{1415168}{1365}\right\}$$

Table[2^n BernoulliB[n, 1], {n, 0, 12}] // TeXForm

$$\left\{1,1,\frac{2}{3},0,-\frac{8}{15},0,\frac{32}{21},0,-\frac{128}{15},0,\frac{2560}{33},0,-\frac{1415168}{1365}\right\}$$

This is a completely natural result. However note how absurd in contrast the following is for n = 1!

Table[2^n BernoulliB[n], {n, 0, 12}] // TeXForm

$$\left\{1,-1,\frac{2}{3},0,-\frac{8}{15},0,\frac{32}{21},0,-\frac{128}{15},0,\frac{2560}{33},0,-\frac{1415168}{1365}\right\}$$

The Genocchi function

Formula 42

The Genocchi function, by definition.

G[s_] := 2^s (Bg[s, 1/2] - Bg[s, 1]);
Clear[s, v]; G[s] // FullSimplify // TeXForm

$$\begin{cases} 2 \left(2^s-1\right) s \zeta (1-s) & s\neq 0 \\ 0 & \text{True} \end{cases}$$

The Genocchi function as the difference between the Bernoulli function and the Bernoulli central function.

Gbc[s_] := 2^s (Bc[s] - B[s]);
Clear[s, v]; Gbc[s] // FullSimplify // TeXForm

$$\begin{cases} 2 \left(2^s-1\right) s \zeta (1-s) & s\neq 0 \\ 1 & \text{True} \end{cases}$$

Table[G[n], {n, 0, 14}]

{0, -1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227}

Plot[G[s], {s, -1, 7},
Epilog -> {PointSize[0.01], Red, Point[Table[{k,G[k]}, {k, 0, 7}]]}]

Formula 43

Genocchi function represented via Bernoulli function.

Gb[s_] := 2 (1 - 2^s) B[s];
Clear[s]; Gb[s] // FullSimplify // TeXForm

$$2 \left(2^s-1\right) s \zeta (1-s)$$

(* Plot[Gb[s], {s, -1, 7}] *)

Formula 44

Generalized Genocchi function.

Gg[s_, x_] := 2^s (Bg[s, x/2] - Bg[s, (x + 1)/2]);

Clear[s, x]; Expand[Gg[s, x]] // FullSimplify // TeXForm

$$\begin{cases} -2^s s \left(\zeta \left(1-s,\frac{x}{2}\right)-\zeta \left(1-s,\frac{x+1}{2}\right)\right) & s\neq 0 \\ 0 & \text{True} \end{cases}$$

Limit[2^s (Bg[s, x/2] - Bg[s, (x + 1)/2]), x -> 1] // TeXForm

$$\begin{cases} 2 \left(2^s-1\right) s \zeta (1-s) & s\neq 0 \\ 0 & \text{True} \end{cases}$$

A226158

Table[Gg[n, 1], {n, 0, 12}]

{0, -1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073}

Formula 45

The Genocchi polynomials.

V := Table[Expand[Simplify[FunctionExpand[
2^n (BernoulliB[n, x/2] - BernoulliB[n, (x + 1)/2])]]], {n, 0, 6}];
MatrixForm[V]

Clear[x]; 
V := Table[Expand[FullSimplify[Gg[n, x]]], {n, 0, 6}]
W := Table[Expand[FullSimplify[Gg[n, x]]], {n, 0, 6}]
U := CoefficientList[W, x]
Grid[{V, W, U}, Frame -> All]

Formula 46

The Genocchi polynomials as difference between the Bernoulli polynomials and the central polynomials.

V := Table[Expand[Simplify[FunctionExpand[
2 (BernoulliB[n, x] - Bcpoly[n, x])]]], {n, 0, 6}];
MatrixForm[V]

w[n_, x_] := Poly[2^n (Bg[n, x/2] - Bg[n, (x + 1)/2])];
W := Table[w[n, x], {n, 0, 6}] 
u[n_, x_] := Poly[Gg[n, x]];
U := Table[u[n, x], {n, 0, 6}] 
Grid[{V, W, U}, Frame -> All]

Table[Expand[FullSimplify[Gg[n, 1]]], {n, 0, 14}]

{0, -1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227}

Table[Expand[FullSimplify[Gg[n, x] / n!]], {n, 3, 7}] 
Plot[%, {x, 0, 1}]

The alternating Bernoulli function

Formula 47

The alternating Riemann zeta function.

atext := MaTeX["\\zeta_{\\text{alt}}(s)"];
btext := MaTeX["\\zeta(s)"];

ζalt[s_] := Zeta[s] (1 - 2^(1 - s));

Plot[{ζalt[s], Zeta[s]}, {s, -8, 12}, PlotRange -> {-3/2, 2}, WorkingPrecision -> 50,
Filling -> {1 -> {2}}, PlotTheme -> {"Thin"},
PlotLegends -> Placed[{atext, btext}, {0.85, 0.15}]]
(* Export["Fig25AlternatingZeta.eps", %] *)

Formula 48

The alternating Bernoulli function, definition.

Balt[s_] := -s ζalt[1 - s];
Balt[0] := 0;

Clear[s]; Balt[s] // FullSimplify // TeXForm

$$\left(2^s-1\right) s \zeta (1-s)$$

atext := MaTeX["B_{\\text{alt}}(s)"];
btext := MaTeX["B(s)"];

Plot[{Balt[s], B[s]}, {s, -7, 7}, PlotTheme -> {"Thin"},
WorkingPrecision -> 50, Filling -> {1 -> {2}}, 
PlotLegends -> Placed[{atext, btext}, {0.85, 0.7}] ]
(* Export["Fig26AlternatingBernoulli.eps", %] *)

Table[Numerator[Balt[n]], {n, 0, 12}]
Table[Denominator[Balt[n]], {n, 0, 12}]

{0, -1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073}

{1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2}

Series[Balt[s], {s, 0, 4}] // TeXForm
% // N // Chop

$$-s \log (2)+s^2 \left(\gamma \log (2)-\frac{\log ^2(2)}{2}\right)+s^3 \left(\gamma _1 \log (2)-\frac{\log ^3(2)}{6}+\frac{1}{2} \gamma \log ^2(2)\right)+\frac{1}{24} s^4 \left(12 \gamma _1 \log ^2(2)+12 \gamma _2 \log (2)-\log ^4(2)+4 \gamma \log ^3(2)\right)+O\left(s^5\right)$$

$$-0.693147 s+0.159869 s^2+0.0326863 s^3+0.00156899 s^4+O\left(s^5\right)$$

Table[(B[n] - Balt[n]) / 2^n, {n, 0, 12}] // TeXForm

$$\left\{1,\frac{1}{2},\frac{1}{6},0,-\frac{1}{30},0,\frac{1}{42},0,-\frac{1}{30},0,\frac{5}{66},0,-\frac{691}{2730}\right\}$$

Formula 49 / 50

The alternating Hurwitz zeta function via Hurwitz zeta.

Limit[2^(-s) (HurwitzZeta[s, x/2] - HurwitzZeta[s, (x + 1)/2]), 
             s -> 1] // TeXForm

$$\frac{1}{2} \left(\psi ^{(0)}\left(\frac{x+1}{2}\right)-\psi ^{(0)}\left(\frac{x}{2}\right)\right)$$

Hζalt[s_, x_] := 2^(-s) (HurwitzZeta[s, x/2] - HurwitzZeta[s, (x + 1)/2]);
Hζalt[1, x_] := (PolyGamma[0, (1 + x)/2] - PolyGamma[0, x/2]) / 2; 
        
Clear[s, x]; Hζalt[s, x] // FullSimplify // TeXForm

$$2^{-s} \left(\zeta \left(s,\frac{x}{2}\right)-\zeta \left(s,\frac{x+1}{2}\right)\right)$$

Table[Hζalt[n, 1], {n, 0, 7}] // N

{0.5, 0.693147, 0.822467, 0.901543, 0.947033, 0.97212, 0.985551, 0.992594}

Plot[{ζalt[s], Hζalt[s, 1]}, {s, -2, 7}, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.3}], {0.9, 0.3}}]]

Formula 51

The alternating Bernoulli polynomials, definition. A333303, A110501, A001469

BHalt[s_, v_] := -s Hζalt[1 - s, v]; 

Clear[s, v]; BHalt[s, v] // FullSimplify // TeXForm

$$2^{s-1} s \left(\zeta \left(1-s,\frac{v+1}{2}\right)-\zeta \left(1-s,\frac{v}{2}\right)\right)$$

Clear[x]; 
Table[Expand[FullSimplify[BHalt[n, x]]], {n, 0, 6}] // MatrixForm

Formula 52

The alternating Bernoulli function via the Bernoulli function.

BBalt[s_] := B[s] (1 - 2^s);
Clear[s]; BBalt[s] // FullSimplify // TeXForm

$$\left(2^s-1\right) s \zeta (1-s)$$

Formula 53

The alternating Bernoulli numbers.

Table[Balt[n], {n, 0, 12}] 
%  // Numerator      (* A036968 *)
%% // Denominator

Formula 53a

The alternating centered Bernoulli numbers. A346464 and A346463

Bcalt1[s_] := Bc[s] (1 - 2^s) 2^s; 
Bcalt2[s_] := Cos[Pi s / 2] B[s] (2^(1 - s) - 1)(2^s - 1) 2^s;
Limit[Bcalt1[s], s -> 0]

Clear[s]; Bcalt1[s] // FullSimplify // TeXForm
Clear[s]; Bcalt2[s] // FullSimplify // TeXForm

0

$$\begin{cases} 0 & s=0 \\ -\left(\left(-3\ 2^s+4^s+2\right) s \zeta (1-s)\right) & \text{True} \end{cases}$$

$$\left(-3\ 2^s+4^s+2\right) s \zeta (1-s) \cos \left(\frac{\pi s}{2}\right)$$

Table[Bcalt1[n], {n, 0, 16, 2}]
Table[Bcalt2[n], {n, 0, 16, 2}]

{0, 1, -7, 93, -2159, 79205, -4243431, 313117357, -30459187423}

{0, 1, 7, 93, 2159, 79205, 4243431, 313117357, 30459187423}

Table[6 QBinomial[2 n, 2, 2] Bg[2 n, 1], {n, 0, 8}]

{0, 1, -7, 93, -2159, 79205, -4243431, 313117357, -30459187423}

a(n) = (4^n - 2)*(4^n - 1) / Clausen(2*n - 1). A346463.

Clausen[n_] := Times @@ (Select[Divisors[n + 1], PrimeQ[# + 1] &] + 1);

Table[((4^n - 2)*(4^n - 1)) / Clausen[2 n - 1], {n, 1, 8}]

{1, 7, 93, 2159, 15841, 6141, 44731051, 8421119}

Table[6 QBinomial[2 n, 2, 2] / Denominator[B[2 n]], {n, 1, 8}]

{1, 7, 93, 2159, 15841, 6141, 44731051, 8421119}

Derivatives of the Bernoulli function

Formula 54

Derivatives of the Bernoulli function.

dB[n_, x_] := If[x == 0, Limit[Derivative[n][B][s], s -> 0], 
               Derivative[n][B][s] /. s -> x]; 

Clear[s, x, n]; dB[n, s] // FullSimplify // TeXForm

$$\text{If}\left[s=0,\underset{s\to 0}{\text{lim}}B^{(n)}(s),B^{(n)}(s)\text{/.}\, s\to s\right]$$

Table[Print[n, " -> ", FullSimplify[dB[n, s]]], {n, 0, 4}] ;

0 -> Piecewise[{{1, s == 0}}, -(s Zeta[1 - s])]
1 -> If[s == 0, Limit[B'[s], s -> 0], B'[s] /. s -> s]
2 -> If[s == 0, Limit[B''[s], s -> 0], B''[s] /. s -> s]
                       (3)               (3)
3 -> If[s == 0, Limit[B   [s], s -> 0], B   [s] /. s -> s]
                       (4)               (4)
4 -> If[s == 0, Limit[B   [s], s -> 0], B   [s] /. s -> s]

dBz[n_, x_] := If[x == 0, Limit[Derivative[n][B][s], s -> 0], 
              ((-1)^n (n Derivative[n - 1][Zeta][1-s] 
                    - s Derivative[n][Zeta][1-s])) /. s -> x];
                    
Clear[s, x, n]; dBz[n, s] // FullSimplify // TeXForm

$$\text{If}\left[s=0,\underset{s\to 0}{\text{lim}}B^{(n)}(s),(-1)^n \left(n \zeta ^{(n-1)}(1-s)-s \zeta ^{(n)}(1-s)\right)\text{/.}\, s\to s\right]$$

Table[Print[n, " -> ", FullSimplify[dBz[n, s]]], {n, 0, 4}] ;

0 -> If[s == 0, 1, -(s Zeta[1 - s])]
1 -> If[s == 0, -EulerGamma, -Zeta[1 - s] + s Zeta'[1 - s]]
2 -> If[s == 0, -2 StieltjesGamma[1], 2 Zeta'[1 - s] - s Zeta''[1 - s]]
                                                               (3)
3 -> If[s == 0, -3 StieltjesGamma[2], -3 Zeta''[1 - s] + s Zeta   [1 - s]]
                                            (3)                (4)
4 -> If[s == 0, -4 StieltjesGamma[3], 4 Zeta   [1 - s] - s Zeta   [1 - s]]

Plot[{dB[1, x], dBz[1, x], dB[2, x], dBz[2, x]}, {x, 0, 6}]

Plot[{-s Zeta[1 - s], 
-Zeta[1 - s] + s Zeta'[1 - s],
2 Zeta'[1 - s] - s Zeta''[1 - s],
-3 Zeta''[1 - s] + s Zeta'''[1 - s]},  
{s, 0, 12}, PlotLegends -> "Expressions"]

(* Export["Fig6BernoulliDerivatives.eps", %] *)

Formula 55

Bernoulli constants as values of a derivative.

Table[dB[n, 0], {n, 0, 6}] // TeXForm
% // N

$$\left\{1,-\gamma ,-2 \gamma _1,-3 \gamma _2,-4 \gamma _3,-5 \gamma _4,-6 \gamma _5\right\}$$

$$\{1.,-0.577216,0.145632,0.0290711,-0.00821534,-0.0116269,-0.00475994\}$$

Table[dBz[n, 0], {n, 0, 6}] // TeXForm
% // N

$$\left\{1,-\gamma ,-2 \gamma _1,-3 \gamma _2,-4 \gamma _3,-5 \gamma _4,-6 \gamma _5\right\}$$

$$\{1.,-0.577216,0.145632,0.0290711,-0.00821534,-0.0116269,-0.00475994\}$$

Table[beta[n], {n, 0, 4}] // Chop

{1.000000000068801, -0.5772156649015378, 0.1456316909673506, 0.02907108957862079, 
 
>   -0.008215337681139509}

dBz[1, 0] == -EulerGamma // TeXForm

$$\text{True}$$

Table[Expand[(n-1)! D[BernoulliB[n, x], x]] /. x -> 1, {n, 1, 16}]

{1, 1, 1, 0, -4, 0, 120, 0, -12096, 0, 3024000, 0, -1576143360, 0, 1525620096000, 0}

Logarithmic derivative and Bernoulli cumulants

For odd integer $n = 3, 5, 7, \ldots $ the value of $\mathcal{L}\!\operatorname{B}(n) $ is undefined.

LdB[s_] := B'[s] / B[s];
LdB[0] := -EulerGamma;

LdG[s_] := Gamma'[s] / Gamma[s];
LdZ[s_] := Zeta'[s] / Zeta[s];

Compare the logarithmic derivative of the Bernoulli function at the even integers with the logarithm of x/π.

Show[Plot[Log[x / Pi], {x, 8, 120}, PlotStyle -> Red],
DiscretePlot[LdB[2 n], {n, 8, 120}, WorkingPrecision -> 70]]

Formula 56

The logarithmic derivative of the Bernoulli function.

LdBz[s_] := 1/s - LdZ[1 - s]; 
LdBz[0] := -EulerGamma; 

Clear[s]; LdBz[s] // FullSimplify // TeXForm

$$\frac{1}{s}-\frac{\zeta '(1-s)}{\zeta (1-s)}$$

Table[{LdBz[s], N[LdBz[s]]}, {s, 0, 9, 2}] // TableForm

Plot[{1/s - LdZ[1 - s], LdB[s]}, {s, -1/2, 7}, 
WorkingPrecision -> 20, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.21, 0.89}], {0.21, 0.89}}]]

Formula 57

Representation of LdB by LdGamma and LdZeta.

ρ[s_] := (1/s) - (Pi/2) Tan[s Pi/2] - Log[2 Pi];  
ρ[s] // TeXForm

$$\frac{1}{s}-\frac{1}{2} \pi \tan \left(\frac{\pi s}{2}\right)-\log (2 \pi )$$

lhs[s_] := LdG[s] + LdZ[s] + ρ[s]; 
Clear[s]; lhs[s] // FullSimplify // TeXForm

$$\frac{\zeta '(s)}{\zeta (s)}+\frac{1}{s}-\frac{1}{2} \pi \tan \left(\frac{\pi s}{2}\right)+\psi ^{(0)}(s)-\log (2 \pi )$$

Plot[{LdZ[s] + LdG[s], LdB[s] - ρ[s]}, {s, -1, 4}, 
WorkingPrecision -> 20, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.9}], {0.9, 0.9}}]]

Formula 58

The coefficients in the series expansion of the logarithmic derivative of the Bernoulli function. The logarithmic polynomials generated by the Bernoulli constants.

K := 3
Series[LdBz[s], {s, 0, K}] 
% /. Table[StieltjesGamma[n] -> -Subscript[β, n+1]/(n+1), {n, 1, K}]
% /. EulerGamma -> -Subscript[β, 1]
CoefficientList[Normal[%], s] Range[0, K]! 
% // Expand // TableForm

Clear[beta, s];
S := Sum[beta[n] s^n / n!, {n, 0, 10}];
beta[0] := 1; 
L := Normal[Series[Log[S], {s, 0, 4}]]; 
Tc := Table[n! Coefficient[L, s, n] , {n, 1, 4}];
Tc // Expand // TableForm // TeXForm

$$\begin{array}{c} \beta (1) \\ \beta (2)-\beta (1)^2 \\ 2 \beta (1)^3-3 \beta (2) \beta (1)+\beta (3) \\ -6 \beta (1)^4+12 \beta (2) \beta (1)^2-4 \beta (3) \beta (1)-3 \beta (2)^2+\beta (4) \\ \end{array}$$

The Hasse-Worpitzky representation

Formula 59

Worpitzky numbers and Fubini polynomials, A163626, A278075.

w[n_, k_] := (-1)^k k! StirlingS2[n + 1, k + 1];
Table[w[n, k], {n, 0, 5}, {k, 0, n}] // MatrixForm

FubiniPoly[n_] := Sum[(-1)^(n-k) StirlingS2[n, k] k! x^k, {k, 0, n}]
Table[FubiniPoly[n], {n, 0, 7}] // MatrixForm

Table[Integrate[FubiniPoly[n], {x, 0, 1}], {n, 0, 12}]

F0 := FubiniPoly[0]; F1 := FubiniPoly[1]; F2 := FubiniPoly[2];
F3 := FubiniPoly[3]; F4 := FubiniPoly[4]; F5 := FubiniPoly[5];
Plot[{F0,F1,F2,F3,F4,F5}, {x, 0, 1}, PlotLegends -> "Expressions",
PlotRange -> {-0.8, 1}]
(* Export["Fig30FubiniPolynomials.eps", %] *)

WorpitzkyPoly[n_] := Sum[w[n, k] x^k, {k, 0, n}]
Table[WorpitzkyPoly[n], {n, 0, 7}] // MatrixForm

Table[Integrate[WorpitzkyPoly[n], {x, 0, 1}], {n, 0, 12}]

WorpitzkyPoly2[n_] := Expand[FubiniPoly[n] /. x -> 1-x]
Table[WorpitzkyPoly2[n], {n, 0, 7}] // MatrixForm

Formula 60

Worpitzky transform.

Wt[n_, a_] := Sum[w[n, k] a[k], {k, 0, n}]

a[n_] := 1 / (n + 1); 
Table[Wt[n, a], {n, 0, 12}] // TeXForm (* Bernoulli numbers *)

$$\left\{1,\frac{1}{2},\frac{1}{6},0,-\frac{1}{30},0,\frac{1}{42},0,-\frac{1}{30},0,\frac{5}{66},0,-\frac{691}{2730}\right\}$$

The generalized Worpitzky transform

Formula 61

Generalized Worpitzky transform.

Clear[a]
Wtg[m_, a_] := Sum[(-1)^n Binomial[m, n] Wt[n, a] x^(m - n), {n, 0, m}];
Clear[m, a]; Wtg[m, a] // FullSimplify // TeXForm

$$\sum _{n=0}^m (-1)^n \binom{m}{n} x^{m-n} \sum _{k=0}^n (-1)^k k! a(k) \mathcal{S}_{n+1}^{(k+1)}$$

Table[Poly[Wtg[n, a]], {n, 0, 4}] // MatrixForm // TeXForm

$$\left( \begin{array}{c} a(0) \\ x a(0)-a(0)+a(1) \\ a(0) x^2+(2 a(1)-2 a(0)) x+a(0)-3 a(1)+2 a(2) \\ a(0) x^3+(3 a(1)-3 a(0)) x^2+(3 a(0)-9 a(1)+6 a(2)) x-a(0)+7 a(1)-12 a(2)+6 a(3) \\ a(0) x^4+(4 a(1)-4 a(0)) x^3+(6 a(0)-18 a(1)+12 a(2)) x^2+(-4 a(0)+28 a(1)-48 a(2)+24 a(3)) x+a(0)-15 a(1)+50 a(2)-60 a(3)+24 a(4) \\ \end{array} \right)$$

a[n_] := 1 / (n + 1);
Table[Wtg[n, a] /. x -> 1, {n, 0, 12}] // TeXForm

$$\left\{1,\frac{1}{2},\frac{1}{6},0,-\frac{1}{30},0,\frac{1}{42},0,-\frac{1}{30},0,\frac{5}{66},0,-\frac{691}{2730}\right\}$$

Table[Wtg[n, HarmonicNumber], {n, 6}] // TeXForm

$$\left\{1,2 x,3 x^2,4 x^3,5 x^4,6 x^5\right\}$$

Formula 62

Generalized Worpitzky transform, alternative form.

Wtga[m_, a_] := Sum[a[n] Sum[(-1)^k Binomial[n, k] (x - k -1)^m, 
                {k, 0, n}], {n, 0, m}];
Clear[m, a]; Wtga[m, a] // FullSimplify // TeXForm

$$\sum _{n=0}^m a(n) \sum _{k=0}^n (-1)^k \binom{n}{k} (-k+x-1)^m$$

Table[Poly[Wtga[n, a]], {n, 0, 4}] // TableForm // TeXForm

$$\begin{array}{c} a(0) \\ x a(0)-a(0)+a(1) \\ a(0) x^2+(2 a(1)-2 a(0)) x+a(0)-3 a(1)+2 a(2) \\ a(0) x^3+(3 a(1)-3 a(0)) x^2+(3 a(0)-9 a(1)+6 a(2)) x-a(0)+7 a(1)-12 a(2)+6 a(3) \\ a(0) x^4+(4 a(1)-4 a(0)) x^3+(6 a(0)-18 a(1)+12 a(2)) x^2+(-4 a(0)+28 a(1)-48 a(2)+24 a(3)) x+a(0)-15 a(1)+50 a(2)-60 a(3)+24 a(4) \\ \end{array}$$

Formula 63

Bernoulli polynomials and Bernoulli numbers from the generalized Worpitzky transform.

Bw[m_] := Sum[(n + 1)^(-1) Sum[(-1)^(m - k) Binomial[n, k] k^m, 
          {k, 0, n}], {n, 0, m}];
Table[Poly[Bw[n]], {n, 0, 12}] // TeXForm

$$\left\{1,\frac{1}{2},\frac{1}{6},0,-\frac{1}{30},0,\frac{1}{42},0,-\frac{1}{30},0,\frac{5}{66},0,-\frac{691}{2730}\right\}$$

a[n_] := 1 / (n + 1);
{Table[Poly[Wtga[n, a]], {n, 0, 5}] // MatrixForm ,
 Table[BernoulliB[n, x], {n, 0, 5}] // MatrixForm }

Examples: a(n) = n + 1 and a(n) = H(n + 1).

a[n_] := n + 1;
Table[Poly[Wtga[n, a]], {n, 0, 5}] // MatrixForm

a[n_] := HarmonicNumber[n + 1];
{Table[Poly[Wtga[n, a]],     {n, 0, 6}] // TableForm, 
Table[Poly[BernoulliB[n,x]], {n, 0, 6}] // TableForm }

The Hasse representation

Formula 64

Hasse's formula.

Bh[s_, v_] := Sum[(n + 1)^(-1) Sum[(-1)^k Binomial[n, k] (k + v)^s, 
              {k, 0, n}], {n, 0, Infinity}];

Clear[s, v, n]; Bh[s, v] // FullSimplify // TeXForm

$$\sum _{n=0}^{\infty } \frac{\sum _{k=0}^n (-1)^k \binom{n}{k} (k+v)^s}{n+1}$$

BhNum[s_, v_] := Sum[(n + 1)^(-1) Sum[(-1)^k Binomial[n, k] (k + v)^s, 
               {k, 0, n}], {n, 0, Ceiling[s]}]; 
               
Table[BhNum[s, 1], {s, 0, 12}] // TeXForm             
Table[2^s BhNum[s, 1/2], {s, 0, 12}] // TeXForm

$$\left\{1,\frac{1}{2},\frac{1}{6},0,-\frac{1}{30},0,\frac{1}{42},0,-\frac{1}{30},0,\frac{5}{66},0,-\frac{691}{2730}\right\}$$

$$\left\{1,0,-\frac{1}{3},0,\frac{7}{15},0,-\frac{31}{21},0,\frac{127}{15},0,-\frac{2555}{33},0,\frac{1414477}{1365}\right\}$$

Show[Plot[BcNum[n], {n, 0, 12}, PlotStyle -> Brown],
DiscretePlot[2^n BhNum[n, 1/2], {n, 0, 12}, WorkingPrecision -> 70]]

Formula 65

The Hasse representation of the central Bernoulli function.

Bhc[s_] := Bh[s, 1/2];
Clear[s]; Bhc[s] // Poly // FullSimplify // TeXForm

$$\sum _{n=0}^{\infty } \frac{\sum _{k=0}^n (-1)^k \left(k+\frac{1}{2}\right)^s \binom{n}{k}}{n+1}$$

Table[2^s BhNum[s, 1/2], {s, 0, 12}]
%  // Numerator   (* even indexed are A001896 *)
%% // Denominator (* A141459, even indexed A001897 *)

Formula 66

For the central Bernoulli numbers equivalent to:

Bhcn[N_] := Sum[(n + 1)^(-1) Sum[(-1)^k Binomial[n, k] (2 k + 1)^N, 
                {k, 0, n}], {n, 0, N}];

Table[Bhcn[s], {s, 0, 12}]
%  // Numerator   
%% // Denominator

Formula 67

Bernoulli constants via Hasse representation

(* Not suited for numerical computation! *)
betah[s_, v_] := Sum[(n + 1)^(-1) Sum[(-1)^k Binomial[n, k] Log[k + v]^s, 
                {k, 0, n}], {n, 0, Infinity}];

Clear[s, v, n]; betah[s, v] // FullSimplify // TeXForm

$$\sum _{n=0}^{\infty } \frac{\sum _{k=0}^n (-1)^k \binom{n}{k} \log ^s(k+v)}{n+1}$$

The functional equation

Formula 68

The tau constant and the tau function.

tau := 2*Pi*I ;
Tau[s_] := tau^(-s) + (-tau)^(-s);
Tau[s] // TeXForm

$$(-2 i \pi )^{-s}+(2 i \pi )^{-s}$$

Plot[Tau[s], {s, -1, 3},  PlotRange -> {-0.1, 3.7}]

Formula 69

ZTG, the ZetaTauGamma product is the reflected Zeta function.

ZTG[s_] := Zeta[s] Tau[s] Gamma[s];
Plot[{ZTG[s], Zeta[1 - s]}, {s, -1, 3}, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.9}], {0.9, 0.9}}]]

Formula 70

ZTF, the -ZetaTauFactorial product is (a representation of) the Bernoulli function.

ZTF[s_] := -Zeta[s] Tau[s] Factorial[s];
Plot[{ZTF[s], B[s]}, {s, -1, 3}, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.8}], {0.9, 0.8}}]]

Plot[{-Zeta[s], \[Tau][s], Factorial[s]}, {s, -1/2, 3}, 
PlotTheme -> {"Thick", "DashedLines"}, PlotRange -> {-9/2, 13/2},
PlotLegends -> Placed["Expressions", {Scaled[{0.25, 0.01}], {0.25, 0.01}}]]  
(* Export["Fig36RiemannDecomposition.eps", %] *)

Formula 71

The functional equation of the Bernoulli function.

Bfeq[s_] := Tau[s] Factorial[s] B[1 - s] / (1 - s);
Plot[{B[s], Bfeq[s]}, {s, -1, 3}, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.8}], {0.9, 0.8}}]]

Formula 72

The symmetric functional equation.

Bsfl[s_] := B[1 - s] (s/2)! / Pi^(s/2);
Bsfr[s_] := B[s] ((1 - s)/2)! / Pi^((1 - s)/2);

Clear[s]; Bsfl[s] // FullSimplify // TeXForm
Clear[s]; Bsfr[s] // FullSimplify // TeXForm

$$\pi ^{-s/2} (s-1) \frac{s}{2}! \zeta (s)$$

$$-\pi ^{\frac{s-1}{2}} s \zeta (1-s) \Gamma \left(\frac{3}{2}-\frac{s}{2}\right)$$

ReImPlot[{Bsfr[s] - Bsfl[s]}, {s, -1, 4}, WorkingPrecision -> 30,
PlotLegends -> Placed["Expressions", {Scaled[{0.5, 0.2}], {0.5, 0.2}}]]

Representation by the Riemann xi function

Formula 73

The Riemann xi (lower case) function, as defined by Landau and, to add to the confusion denoted by Mathematica as RiemannXi, although it is not the Riemann Xi (upper case) function, as defined by Landau with upper case and denoted by Riemann with xi (lower case).

xi[s_] := (s/2)! Pi^(-s/2) (s - 1) Zeta[s];
xi[s]

FullSimplify[RiemannXi[u] - xi[u]] // TeXForm

Formula 74

The Bernoulli function in terms of the Riemann xi function.

Bxi[s_] := If[n == 1, 1/2, (Pi^((1 - s) / 2) / ((1 - s) / 2)!) xi[s]];
Clear[s, v, n]; Bxi[s] // FullSimplify // TeXForm

$$\begin{cases} \frac{1}{2} & n=1 \\ -(2 \pi )^{-s} \zeta (s) \sin (\pi s) \csc \left(\frac{\pi s}{2}\right) \Gamma (s+1) & \text{True} \end{cases}$$

Table[Bxi[n], {n, 0, 8}] // N
Table[B[n],   {n, 0, 8}] // N

{1., 0.5, 0.166667, 0., -0.0333333, 0., 0.0238095, 0., -0.0333333}

{1., 0.5, 0.166667, 0., -0.0333333, 0., 0.0238095, 0., -0.0333333}

Formula 75

The Bernoulli function in terms of the Riemann xi function.

Bxiref[s_] := (Pi^(s/2) / (s/2)!) RiemannXi[s];

Table[Bxiref[n], {n, 0, 8}] // N
Table[B[1 - n], {n, 0, 8}] // N

{0.5, 1., 1.64493, 2.40411, 3.24697, 4.14771, 5.08672, 6.0501, 7.02854}

{0.5, 1., 1.64493, 2.40411, 3.24697, 4.14771, 5.08672, 6.0501, 7.02854}

Formula 76

The Basel problem. The solution is: The Bernoulli function at s = -1.

{B[-1], Pi RiemannXi[-1], Pi RiemannXi[2]} // TeXForm

$$\left\{\frac{\pi ^2}{6},\frac{\pi ^2}{6},\frac{\pi ^2}{6}\right\}$$

The Hadamar decomposition

Formula 77 & 78

Hadamard's infinite product expansion of the zeta function.

H[s_] := Zeta[s] (s - 1) (s/2)! / Pi^(s/2);
H[1] := 1/2;
Clear[s]; H[s] // FullSimplify // TeXForm

h[s_] := With[{sigma = (1 - s)/2}, Pi^sigma / sigma!];
Clear[s]; h[s] // FullSimplify // TeXForm

$$\pi ^{-s/2} (s-1) \frac{s}{2}! \zeta (s)$$

$$\frac{\pi ^{\frac{1}{2}-\frac{s}{2}}}{\Gamma \left(\frac{3}{2}-\frac{s}{2}\right)}$$

Formula 79

Hadamard decomposition of the Bernoulli function.

Plot[{H[s] h[s], B[s], RiemannXi[s] h[s] }, {s, 0, 8},  
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.8}], {0.9, 0.8}}]]

Formula 80

Jensen's formula for the Hadamard product and the Riemann xi function.

Clear[a, b]
JensenInt[s_, a_, b_] := 2 (s/2)! Pi^(1 - s/2) NIntegrate[Re[(1/2 + I x)^(1 - s) / 
         (Exp[Pi x] + Exp[-Pi x])^2], {x, a, b}]; 
         
HoldForm[2 (s/2)! Pi^(1 - s/2) Integrate[Re[(1/2 + I x)^(1 - s)
         / (Exp[Pi x] + Exp[-Pi x])^2], {x, -Infinity, Infinity}]] // TeXForm

$$2 \frac{s}{2}! \pi ^{1-\frac{s}{2}} \int_{-\infty }^{\infty } \Re\left(\frac{\left(\frac{1}{2}+i x\right)^{1-s}}{(\exp (\pi x)+\exp (-\pi x))^2}\right) \, dx$$

(* The splitting helps to evaluate the integral numerically. *)

HJ[s_] := JensenInt[s, -Infinity, 1] + JensenInt[s, 1, Infinity];          

Table[HJ[s], {s, 1, 4}]
Table[H[s], {s, 1, 4}] // N
Table[RiemannXi[s], {s, 1, 4}] // N

{0.5, 0.523599, 0.57394, 0.657974}

{0.5, 0.523599, 0.57394, 0.657974}

{0.5, 0.523599, 0.57394, 0.657974}

ComplexExpand[Re[(1/2 + I z)^(1 - s)/(E^(-(Pi z)) + E^(Pi z))^2]]

The generalized Euler function

Formula 81

Generalized Euler function.

Eg[s_, v_] := -Gg[s + 1, v] / (s + 1); 
Eg[-1, v_] := Log[4]; 

Clear[s, v]; Eg[s, v] // FullSimplify // TeXForm

$$\begin{cases} 2^{s+1} \left(\zeta \left(-s,\frac{v}{2}\right)-\zeta \left(-s,\frac{v+1}{2}\right)\right) & s\neq -1 \\ 0 & \text{True} \end{cases}$$

Test (see also A002425).

Table[Eg[n, 1], {n, 0, 13}] // SeqGrid
Table[2^n Eg[n, 1], {n, 0, 11}] // SeqGrid

Euler polynomials

V := Table[Expand[FullSimplify[Eg[n, x]]], {n, 1, 6}]
W := Table[EulerE[n, x], {n, 1, 6}]
Grid[{V, W}, Frame -> All]

Plot[V, {x, -1, 3/2}, PlotRange -> {-1, 1/2}]

The Euler tangent function

Formula 82

Euler tangent function, definition.

Et[s_] := 2^s Eg[s, 1];
Et[-1] := Log[2];

Clear[s]; Et[s] // FullSimplify // TeXForm

$$\begin{cases} -2^{s+1} \left(2^{s+1}-1\right) \zeta (-s) & s\neq -1 \\ 0 & \text{True} \end{cases}$$

ReImPlot[Et[s], {s, -1, 6}, PlotRange -> {-3, 23}, 
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.8}], {0.3, 0.8}}]]

Formula 82a

Euler tangent function, via polylogarithm.

EtLi[s_] := -2 Re[PolyLog[-s, I]];

Clear[s]; EtLi[s] // FullSimplify // TeXForm

$$-2 \Re(\text{Li}_{-s}(i))$$

Test (Note that -2 Re[-Log[1 - I]] = Log[2]).

Table[EtLi[n], {n, 0, 11}]

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792}

(* Plot[EtLi[s], {s, -1, 6}, PlotRange -> {-3, 23}] *)

Formula 83

A155585

Table[Et[n], {n, 0, 11}]

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792}

Formula 84

Euler tangent function, via Bernoulli function.

EtB[s_] := B[s + 1] (4^(s + 1) - 2^(s + 1)) / (s + 1);
EtB[-1] := Log[2]; 
           
Clear[s]; EtB[s] // FullSimplify // TeXForm

$$-2^{s+1} \left(2^{s+1}-1\right) \zeta (-s)$$

Table[2^n Eg[n, 1], {n, 0, 11}]
Table[Et[n], {n, 0, 11}]
Table[EtB[n], {n, 0, 11}]

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792}

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792}

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792}

(* Plot[EtB[s], {s, -1, 6}, PlotRange -> {-3, 23}] *)

Formula 85

Connection to the Eulerian numbers, A173018.

EulerA[n_ /; n >= 0, 0] = 1; EulerA[n_, k_] /; k < 0 || k > n = 0;
EulerA[n_, k_] := EulerA[n, k] = (n-k)*EulerA[n-1, k-1] + (k+1)*EulerA[n-1, k]; 
Table[EulerA[n, k], {n, 0, 7}, {k, 0, n}] // TableForm

Formula 86

$ 2^n E_n(1) = A_n(-1)$

EulerApoly[n_] := Sum[EulerA[n, k] x^k, {k, 0, n}];
Table[EulerApoly[n] /. x -> -1, {n, 0, 9}]

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936}

Formula 87

Stirling-Fubini type polynomials.

sf[n_] := Sum[(-2)^(n - k) StirlingS2[n, k] k! x^k, {k, 0, n}];

Table[sf[n] /. x -> 1, {n, 0, 9}]

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936}

A122704

Table[(-1)^n sf[n] /. x -> -1, {n, 0, 9}]

{1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216}

A122704[n_] := (-2)^(n + 1) PolyLog[-n, 3] / 3;
Table[A122704[n], {n, 0, 9}]

{1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216}

The Euler secant function

Formula 88

Euler secant function, definition.

Es[s_] := If[1 + s == 0, Pi/2, 2^s Eg[s, 1/2]];

Clear[s]; Es[s] // FullSimplify // TeXForm

$$\text{If}\left[s+1=0,\frac{\pi }{2},2^s \text{Eg}\left(s,\frac{1}{2}\right)\right]$$

Fake traditional notation since 'E' is protected.

\[CapitalEpsilon][s_] := Es[s];

ReImPlot[Es[s], {s, -1, 6}, PlotRange -> {-3/2, 13/2},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.8}], {0.3, 0.8}}]]

Formula 89

Euler (secant) numbers.

Table[FullSimplify[Es[n]], {n, 0, 12}] // SeqGrid

Table[FullSimplify[AbsEs[n]], {n, 0, 12}] // SeqGrid

Formula 90

Euler secant function, via generalized Bernoulli.

EsB1[s_] := 2 * 4^s (Bg[s + 1, 3/4] - Bg[s + 1, 1/4]) / (s + 1);
EsB1[-1] := Pi / 2; 
           
Clear[s]; EsB1[s] // FullSimplify // TeXForm

$$\begin{cases} 2^{2 s+1} \left(\zeta \left(-s,\frac{1}{4}\right)-\zeta \left(-s,\frac{3}{4}\right)\right) & s\neq -1 \\ 0 & \text{True} \end{cases}$$

EsB[s_] := (2^(s + 1) Bg[s + 1, 1/2] - 4^(s + 1) Bg[s + 1, 1/4]) / (s + 1);
EsB[-1] := Pi / 2; 
           
Clear[s]; EsB[s] // FullSimplify // TeXForm

$$\begin{cases} 4^{s+1} \zeta \left(-s,\frac{1}{4}\right)+2 \left(2^s-1\right) \zeta (-s) & s\neq -1 \\ \text{Indeterminate} & \text{True} \end{cases}$$

Table[FullSimplify[EsB[n]],  {n, 0, 12}] 
Table[FullSimplify[EsB1[n]], {n, 0, 12}]

{1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765}

{1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765}

Clear[s]; FullSimplify[EsB[s] - EsB1[s]]

Plot[{EsB[s], EsB1[s]}, {s, -1, 6}, PlotRange -> {-3/2, 13/2},
PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.8}], {0.3, 0.8}}]]

Formula 90a

Euler secant function, via polylogarithm.

EsLi[s_] := 2 Im[PolyLog[-s, I]];
Clear[s]; EsLi[s] // FullSimplify // TeXForm

$$2 \Im(\text{Li}_{-s}(i))$$

Table[EsLi[n], {n, 0, 10}]   (* 2 Im[-Log[1-I]] = Pi/2 *)

{1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521}

(* Plot[EsLi[s], {s, -1, 6}, PlotRange -> {-3/2, 13/2}] *)

atext := MaTeX["E_{\\tau}(s)"];
btext := MaTeX["E_{\\sigma}(s)"];

Etan[s_] := Et[s];
Esec[s_] := Es[s];
Plot[{Etan[s], Esec[s]}, {s, -1, 6}, PlotRange -> {-9, 23}, 
Filling -> {1 -> {2}}, WorkingPrecision -> 60,
PlotLegends -> Placed[{atext, btext}, {0.35, 0.8}]]  
(* Export["Fig23EulerTanSec.eps", %] *)

Formula 91

Even indexed classical Euler numbers via an Jensen integral.

EInt[s0_, z0_, prec_] := Module[{s, z}, 
   {s, z} = SetPrecision[{s0, z0}, prec $MachinePrecision];
   Block[{$MinPrecision = prec $MachinePrecision, 
   $MaxPrecision = prec $MachinePrecision},
   ((4 z I + 1)^(s + 1) - (4 z I - 1)^(s + 1))/(Exp[-Pi z] + Exp[Pi z])^2]];
       
JEInt[s0_, prec_] := Module[{s}, 
   {s} = SetPrecision[{s0}, prec $MachinePrecision];
   ((2 Pi ) / (s + 1)) 
   NIntegrate[EInt[s, z, prec], {z, 0, Infinity}, 
   WorkingPrecision -> prec $MachinePrecision ]];

A346838

Table[JEInt[2 n, 2], {n, 0, 6}] // Chop

{1.0000000000000000000000000000000, -1.0000000000000000000000000000000, 
 
>   5.000000000000000000000000000000, -61.00000000000000000000000000000, 
 
>   1385.0000000000000000000000000000, -50521.00000000000000000000000000, 
 
                                        6
>   2.7027650000000000000000000000000 10 }

Euler Zeta numbers

Formula 92

Euler Zeta numbers.

{1, 1, 1/2, 1/3, 5/24, 2/15, 61/720, 17/315, 277/8064, 62/2835} // TeXForm

$$\left\{1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24},\frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835}\right\}$$

The Bernoulli secant function

Formula 93

The Bernoulli secant function.

Bsec[s_] := (2^(s - 1) / (2^s - 1))(Bg[s, 3/4] - Bg[s, 1/4]);

Clear[s]; Bsec[s] // FullSimplify // TeXForm

$$\begin{cases} \frac{2^{s-1} s \left(\zeta \left(1-s,\frac{1}{4}\right)-\zeta \left(1-s,\frac{3}{4}\right)\right)}{2^s-1} & s\neq 0 \\ 0 & \text{True} \end{cases}$$

Formula 94

A160143, A193476

Table[FullSimplify[Bsec[n]], {n, 1, 10}]
%  // Numerator   
%% // Denominator

atext := MaTeX["B_{\\tau}(s)"];
btext := MaTeX["B_{\\sigma}(s)"];
Btan[s_] := B[s];
Plot[{Btan[s], Bsec[s]}, {s, 1, 8}, 
PlotRange -> {-0.1, 1/2},  WorkingPrecision -> 50, Filling -> {1 -> {2}},
PlotLegends -> Placed[{atext, btext}, {0.8, 0.6}] ]  
(* Export["Fig24BernoulliTanSec.eps", %] *)

Formula 95

The Bernoulli secant numbers represented by the Euler secant numbers.

BsecE[n_] := (-1)^(n - 1) (n / (4^n - 2^n)) Es[n - 1]; 

Clear[n]; BsecE[n] // FullSimplify // TeXForm

$$\frac{(-1)^{n+1} n \text{If}\left[n=0,\frac{\pi }{2},2^{n-1} \text{Eg}\left(n-1,\frac{1}{2}\right)\right]}{4^n-2^n}$$

Table[FullSimplify[BsecE[n]], {n, 1, 11}] // TeXForm

$$\left\{\frac{1}{2},0,-\frac{3}{56},0,\frac{25}{992},0,-\frac{427}{16256},0,\frac{12465}{261632},0,-\frac{555731}{4192256}\right\}$$

Formula 96

The Bernoulli secant function via the polylogarithm.

BsecP[s_] := ((2 s) / (4^s - 2^s)) Im[PolyLog[1 - s, I]];
BsecP[0] := (2 Pi) / Log[16];

Clear[n]; BsecP[s] // FullSimplify // TeXForm

$$\frac{2 s \Im(\text{Li}_{1-s}(i))}{4^s-2^s}$$

Table[FullSimplify[BsecP[n]], {n, 1, 11}] // TeXForm

$$\left\{\frac{1}{2},0,-\frac{3}{56},0,\frac{25}{992},0,-\frac{427}{16256},0,\frac{12465}{261632},0,-\frac{555731}{4192256}\right\}$$

Plot[{BsecP[s], Bsec[s]}, {s, -1, 6}, 
PlotTheme -> {"Thick", "DashedLines"}, WorkingPrecision -> 30, 
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.9}], {0.9, 0.9}}]]

The extended Bernoulli function

Formula 97

Extended Zeta function.

Zext[s_] := Zeta[s] - (Zeta[s, 3/4] - Zeta[s, 1/4]) / (2^s - 2);
Zext[s] // TeXForm

$$\zeta (s)-\frac{\zeta \left(s,\frac{3}{4}\right)-\zeta \left(s,\frac{1}{4}\right)}{2^s-2}$$

Formula 98

Extended Bernoulli function, definition via generalized Zeta.

Bext[s_] := -s Zext[1 - s];
Bext[0] := 1 + Pi/Log[4];

Clear[s]; Bext[s] // FullSimplify // TeXForm

$$\frac{s \left(\zeta \left(1-s,\frac{3}{4}\right)-\zeta \left(1-s,\frac{1}{4}\right)\right)}{2^{1-s}-2}-s \zeta (1-s)$$

(* N[Pi + PolyGamma[0, 1/4] - PolyGamma[0, 3/4], 20] *)
Series[Bext[s], {s, 0, 7}] // Normal // N

CoefficientList[%, s] // SeqGrid

Formula 99

Extended Bernoulli function via generalized Bernoulli, B(s, v).

BextB[s_] := B[s] + (2^(s - 1) / (2^s - 1))(Bg[s, 3/4] - Bg[s, 1/4]);
BextB[0]  := 1 + Pi/Log[4]; 

Clear[s]; BextB[s] // FullSimplify // TeXForm

$$\begin{cases} 0 & s=0 \\ \frac{s \left(2^s \zeta \left(1-s,\frac{1}{4}\right)-2^s \zeta \left(1-s,\frac{3}{4}\right)-2 \left(2^s-1\right) \zeta (1-s)\right)}{2 \left(2^s-1\right)} & \text{True} \end{cases}$$

Plot[{BextB[s], B[s]}, {s, 2, 8}, PlotRange -> {-0.1, 0.2},
WorkingPrecision -> 40, Filling -> {1 -> {2}},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.9}], {0.9, 0.9}}]]

Formula 100

The extended Bernoulli function is the sum of the Bernoulli function and the Bernoulli secant function.

Clear[s]; Btan[s] + Bsec[s] // FullSimplify // TeXForm

$$\begin{cases} 0 & s=0 \\ \frac{s \left(2^s \zeta \left(1-s,\frac{1}{4}\right)-2^s \zeta \left(1-s,\frac{3}{4}\right)-2 \left(2^s-1\right) \zeta (1-s)\right)}{2 \left(2^s-1\right)} & \text{True} \end{cases}$$

Plot[{Bext[s], Btan[s], Bsec[s]}, {s, -1, 6}, 
PlotTheme -> {"Thick", "DashedLines"}, WorkingPrecision -> 30, 
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.9}], {0.9, 0.9}}]]

Plot[{Bext[s], Btan[s] + Bsec[s]}, {s, -1, 6}, 
PlotTheme -> {"Thick", "DashedLines"}, WorkingPrecision -> 30, 
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.9}], {0.9, 0.9}}]]

Formula 101

The extended Bernoulli numbers are the values of the extended Bernoulli function for integers n ≥ 1.

Bextn[n_] := FullSimplify[BextB[n]];

Table[Bextn[n], {n, 1, 10}]
%  // Numerator   // Print
%% // Denominator // Print

{1, 1, -3, -1, 25, 1, -427, -1, 12465, 5}
{1, 6, 56, 30, 992, 42, 16256, 30, 261632, 66}

The extended Euler Function

Formula 102

Extended Euler function via extended Bernoulli.

Eext[s_] := (4^(s + 1) - 2^(s + 1)) (Bext[s + 1] / (s + 1));
Eext[-1] := Pi / 2 + Log[2];

Clear[s]; Eext[s] // FullSimplify // TeXForm

$$-2^{s+1} \left(\left(2^{s+1}-1\right) \zeta (-s)+2^s \left(\zeta \left(-s,\frac{3}{4}\right)-\zeta \left(-s,\frac{1}{4}\right)\right)\right)$$

Series[Eext[s], {s, 0, 7}] // Normal // N 
CoefficientList[%, s] // SeqGrid

Table[FullSimplify[Eext[n]], {n, 0, 12}]

{2, 1, -1, -2, 5, 16, -61, -272, 1385, 7936, -50521, -353792, 2702765}

Formula 103

Extended Euler function via extended Zeta.

EextZ[s_] := Zext[-s] (2^(1+s) - 4^(1 + s));
EextZ[-1] := Pi/2 + Log[2]; 
    
Clear[s]; EextZ[s] // FullSimplify // TeXForm

$$-2^{s+1} \left(\left(2^{s+1}-1\right) \zeta (-s)+2^s \left(\zeta \left(-s,\frac{3}{4}\right)-\zeta \left(-s,\frac{1}{4}\right)\right)\right)$$

Formula 104

The extended Euler function is the sum of the Euler secant and the Euler tangent function.

Est[s_] := Es[s] + Et[s];

Clear[s]; Est[s] // FullSimplify // TeXForm

$$\begin{cases} \frac{\pi }{2} & s=-1 \\ 2^{s+1} \left(2^s \left(\zeta \left(-s,\frac{1}{4}\right)-\zeta \left(-s,\frac{3}{4}\right)-2 \zeta (-s)\right)+\zeta (-s)\right) & \text{True} \end{cases}$$

Plot[{Es[s] + Et[s], Eext[s]}, {s, -1, 6}, 
PlotRange -> {-9, 17}, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.9}], {0.3, 0.9}}]]

The scaled extended Euler function.

Plot[Eext[s] / s!, {s, 6, 18}, WorkingPrecision -> 60 ]

Formula 105

See also OEIS A163982.

Table[FullSimplify[Eext[n]], {n, 0, 11}]
Table[FullSimplify[EextZ[n]], {n, 0, 11}]
Table[FullSimplify[Est[n]], {n, 0, 11}]

{2, 1, -1, -2, 5, 16, -61, -272, 1385, 7936, -50521, -353792}

{2, 1, -1, -2, 5, 16, -61, -272, 1385, 7936, -50521, -353792}

{2, 1, -1, -2, 5, 16, -61, -272, 1385, 7936, -50521, -353792}

Formula 106

Euler extended function via generalized Bernoulli.

EextBg[-1] := Pi/2 + Log[2];
EextBg[s_] := (((4^(s+1) - 2^(s+2) + 2) B[s + 1] 
               - 4^(s + 1) Bg[s + 1, 1/4]) / (s + 1));
    
Clear[s]; EextBg[s] // FullSimplify // TeXForm

$$\begin{cases} 4^{s+1} \zeta \left(-s,\frac{1}{4}\right)-2 \left(-2^{s+1}+2^{2 s+1}+1\right) \zeta (-s) & s\neq -1 \\ \text{ComplexInfinity} & \text{True} \end{cases}$$

Table[FullSimplify[EextBg[n]], {n, 0, 11}]

{2, 1, -1, -2, 5, 16, -61, -272, 1385, 7936, -50521, -353792}

Plot[{EextBg[s], Eext[s]},{s, -1, 6}, PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.9}], {0.3, 0.9}}]]

Plot[{Esec[s], Etan[s], Eext[s]},{s, -1, 6}, PlotTheme -> {"Thick", "DashedLines"},
WorkingPrecision -> 30, PlotRange -> {-10, 20},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.9}], {0.3, 0.9}}]]

Formula 107

Jensen integral representation of the extended Euler numbers.
The polynomials are the numerators in the integral. See A342317

Ef[s0_, z0_, prec_] := Module[{s, z}, 
   {s, z} = SetPrecision[{s0, z0}, prec $MachinePrecision];
   Block[{$MinPrecision = prec $MachinePrecision, 
   $MaxPrecision = prec $MachinePrecision},
   (((1 + 4 z I)^(s + 1) + (1 - 2^(-2 s - 1))(2 + 4 z I)^(s + 1))
     / (Exp[-Pi z] + Exp[Pi z])^2 )]];

JEfIntegral[s0_, prec_] := Module[{s}, 
     {s} = SetPrecision[{s0}, prec $MachinePrecision];
     ((2 Pi) / (s + 1)) NIntegrate[Ef[s, z, prec], {z, -Infinity, Infinity},
     WorkingPrecision -> prec $MachinePrecision ]];

Table[Round[JEfIntegral[n, 2]], {n, 0, 12}]

{2, 1, -1, -2, 5, 16, -61, -272, 1385, 7936, -50521, -353792, 2702765}

Plot[{JEfIntegral[s, 1], Eext[s]}, {s, -1, 6}, 
PlotTheme -> {"Thick", "DashedLines"},
WorkingPrecision -> 30, PlotRange -> {-10, 20},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.9}], {0.3, 0.9}}]]

The André function

Formula 108, 109

The unsigned Euler functions.

AbsEt[s_] := Sin[Pi s / 2] Et[s]; 
AbsEt[-1] := -Log[2];
Clear[s]; AbsEt[s] // FullSimplify // TeXForm

AbsEs[s_] := Cos[Pi s / 2] Es[s]
Clear[s]; AbsEs[s] // FullSimplify // TeXForm

$$\begin{cases} -2^{s+1} \left(2^{s+1}-1\right) \zeta (-s) \sin \left(\frac{\pi s}{2}\right) & s\neq -1 \\ 0 & \text{True} \end{cases}$$

$$\cos \left(\frac{\pi s}{2}\right) \text{If}\left[s+1=0,\frac{\pi }{2},2^s \text{Eg}\left(s,\frac{1}{2}\right)\right]$$

Euler secant function, unsigned version.

AbsEs[s_] := Cos[Pi s / 2] Es[s]
           
Clear[s]; AbsEs[s] // FullSimplify // TeXForm

$$\cos \left(\frac{\pi s}{2}\right) \text{If}\left[s+1=0,\frac{\pi }{2},2^s \text{Eg}\left(s,\frac{1}{2}\right)\right]$$

Table[FullSimplify[AbsEs[n]], {n, 0, 11}]

{1, 0, 1, 0, 5, 0, 61, 0, 1385, 0, 50521, 0}

ReImPlot[Cos[Pi s / 2] Es[s], {s, -1, 6}, PlotRange -> {-3/2, 13/2},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.8}], {0.3, 0.8}}]]

Euler tangent function, unsigned version.

AbsEt[s_] := Cos[Pi (s - 1) / 2] Et[s]; 
AbsEt[-1] := -Log[2];
           
Clear[s]; AbsEt[s] // FullSimplify // TeXForm

$$\begin{cases} -2^{s+1} \left(2^{s+1}-1\right) \zeta (-s) \sin \left(\frac{\pi s}{2}\right) & s\neq -1 \\ 0 & \text{True} \end{cases}$$

Test, compare OEIS A009006.

Table[AbsEt[n], {n, 0, 11}]

{0, 1, 0, 2, 0, 16, 0, 272, 0, 7936, 0, 353792}

ReImPlot[Sin[Pi s / 2] Et[s], {s, -1, 6}, PlotRange -> {-1, 20},
PlotLegends -> Placed["Expressions", {Scaled[{0.3, 0.8}], {0.3, 0.8}}]]

Formula 110

The André function.

A[s_] := AbsEt[s] + AbsEs[s];
Clear[s]; A[s] // FullSimplify // TeXForm

$$\begin{cases} 0 & s=-1 \\ 2^{s+1} \left(2^s \left(\zeta \left(-s,\frac{1}{4}\right)-\zeta \left(-s,\frac{3}{4}\right)\right) \cos \left(\frac{\pi s}{2}\right)-\left(2^{s+1}-1\right) \zeta (-s) \sin \left(\frac{\pi s}{2}\right)\right) & \text{True} \end{cases}$$

atext := MaTeX["|E|_{\\tau}(s)"];
btext := MaTeX["|E|_{\\sigma}(s)"];
ctext := MaTeX["A(s)\, = \, |E|_{\\tau}(s) + |E|_{\\sigma}(s)"];

Plot[{AbsEt[s], AbsEs[s], A[s]}, {s, -1, 6}, PlotRange -> {-1, 25}, 
Filling -> {1 -> {2}}, WorkingPrecision -> 60,
PlotLegends -> Placed[{atext, btext, ctext}, {0.4, 0.7}],
Epilog -> {PointSize[0.01], Red, Point[Table[{k, A[k]}, {k, 0, 5}]]}]  
(* Export["Fig37AndreFunction.eps", %] *)

Formula 111

The André function via the polylogarithm.

Apl[s_] := (-I)^(s+1) PolyLog[-s, I] + I^(s+1) PolyLog[-s, -I];
Clear[s]; Apl[s] // FullSimplify // TeXForm

$$i^{s+1} \text{Li}_{-s}(-i)+(-i)^{s+1} \text{Li}_{-s}(i)$$

Plot[{A[s], Apl[s]}, {s, -5, 5},
PlotTheme -> {"Thick", "DashedLines"},
WorkingPrecision -> 30, PlotRange -> {-10, 20},
PlotLegends -> Placed["Expressions", {Scaled[{0.1, 0.9}], {0.1, 0.9}}]]

Formula 112

The André numbers.

Table[FullSimplify[A[n]], {n, 0, 11}]

{1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792}

Formula 113

The André numbers for positive integers.

Prepend[Table[2 (-I)^(s + 1) PolyLog[-s, I], {s, 1, 11}], 1]

{1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792}

Plot[{A[s], 2 Re[(-I)^(s + 1) PolyLog[-s, I]]}, {s, -5, 5},
PlotTheme -> {"Thick", "DashedLines"},
WorkingPrecision -> 30, PlotRange -> {-10, 20},
PlotLegends -> Placed["Expressions", {Scaled[{0, 0.9}], {0, 0.9}}]]

Formula 114

Euler zeta numbers. A099612 / A099617

Table[Apl[n] / n!, {n, 0, 10}] 
%  // Numerator   // Print 
%% // Denominator // Print

{1, 1, 1, 1, 5, 2, 61, 17, 277, 62, 50521}
{1, 1, 2, 3, 24, 15, 720, 315, 8064, 2835, 3628800}

Formula 115

The signed André function.

As[s_] := I (PolyLog[-s, -I] - Exp[I Pi s] PolyLog[-s, I]) / Exp[I Pi s / 2];
Clear[s]; As[s] // FullSimplify // TeXForm

$$i e^{-\frac{1}{2} i \pi s} \left(\text{Li}_{-s}(-i)-e^{i \pi s} \text{Li}_{-s}(i)\right)$$

atext := MaTeX["A(s)"];
btext := MaTeX["A^{\\ast}(s)"];

Plot[{A[s], As[s]},{s, -1.2, 4.6}, 
PlotRange -> {-4, 9}, WorkingPrecision -> 30,
Epilog -> {PointSize[0.01], Red, Point[Table[{k, (-1)^k A[k]}, {k, 0, 4}]]},
PlotLegends -> Placed[{atext, btext}, {0.4, 0.8}]]
(* Export["Fig31AndreFunctions.eps", %] *)

The scaled André functions.

atext := MaTeX["\\frac{A(s)}{s!}"];
btext := MaTeX["\\frac{A^{\\ast}(s)}{s!}"];

Plot[{A[s] / s!, As[s] /s! }, {s, -4, 6},
PlotRange -> {-2.0, 1.5}, Filling -> {1 -> {2}},
PlotLegends -> Placed[{atext, btext}, {0.85, 0.2}],
Epilog -> {PointSize[0.01], Red, Point[Table[{k, A[k] / k!}, {k, 0, 6}]]} ]
(* Export["Fig35ScaledAndreFunctions.eps", %]  *)

Formula 116

The signed Andre numbers. A346838, A346839.

Table[As[n], {n, 0, 11}] // FullSimplify

{1, -1, 1, -2, 5, -16, 61, -272, 1385, -7936, 50521, -353792}

Table[As[n] / n!, {n, 0, 29}] // FullSimplify // Accumulate // N

{1., 0., 0.5, 0.166667, 0.375, 0.241667, 0.326389, 0.272421, 0.306771, 0.284901, 
 
>   0.298824, 0.28996, 0.295603, 0.292011, 0.294298, 0.292842, 0.293769, 0.293178, 
 
>   0.293554, 0.293315, 0.293467, 0.29337, 0.293432, 0.293393, 0.293418, 0.293402, 
 
>   0.293412, 0.293405, 0.29341, 0.293407}

The Seki function

Formula 117, 118

The unsigned Bernoulli functions.

bs[s_] := Sin[Pi s / 2] Bsec[s];
bt[s_] := Cos[Pi s / 2] Btan[s];
Table[FullSimplify[bs[s] - bt[s]], {s, 1, 12}]

Formula 119

The Seki function.

Seki[s_] := If[s == 0, -1, bs[s] - bt[s]];
Clear[s]; Seki[s] // FullSimplify // TeXForm

$$\text{If}[s=0,-1,\text{bs}(s)-\text{bt}(s)]$$

atext := MaTeX["|B|_{\\tau}(s)"];
btext := MaTeX["|B|_{\\sigma}(s)"];
ctext := MaTeX["S(s)\, = \, |B|_{\\sigma}(s) - |B|_{\\tau}(s)"];

Plot[{-bt[s], bs[s], Seki[s]}, {s, -0.4, 14}, 
Filling -> {1 -> {2}}, WorkingPrecision -> 60, PlotRange -> {-0.4, 1.012},
PlotLegends -> Placed[{atext, btext, ctext}, {0.47, 0.8}],
Epilog -> {PointSize[0.01], Red, Point[Table[{k, Seki[k]}, {k, 1, 13}]]}]  
(* Export["Fig38SekiFunction.eps", %] *)

Formula 120

The Seki function via the André function.

SekiA[s_] := s Apl[s-1] / (4^s - 2^s);
SekiA[0]  := -1; 
SekiA[-1] := -8 Catalan; 

Clear[s]; SekiA[s] // FullSimplify // TeXForm

$$\frac{s \left(i^s \text{Li}_{1-s}(-i)+(-i)^s \text{Li}_{1-s}(i)\right)}{4^s-2^s}$$

Formula 121

The Seki function via the polylogarithm.

SekiP[s_] := s (I^s PolyLog[1-s, -I] + (-I)^s PolyLog[1-s, I]) / (4^s - 2^s);
SekiP[0] := -1; 
Clear[s]; SekiP[s] // FullSimplify // TeXForm

$$\frac{s \left(i^s \text{Li}_{1-s}(-i)+(-i)^s \text{Li}_{1-s}(i)\right)}{4^s-2^s}$$

SekiP2[s_] := Re[(2 s (-I)^s PolyLog[1-s, I]) / (4^s - 2^s)];
SekiP2[0]  := -1; 

Clear[s]; SekiP2[s] // FullSimplify // TeXForm

$$2 \Re\left(\frac{\left(-\frac{i}{2}\right)^s s \text{Li}_{1-s}(i)}{2^s-1}\right)$$

Table[Seki[n], {n, 0, 12}] // FullSimplify
Table[SekiA[n], {n, 0, 12}] 
Table[SekiP[n], {n, 0, 12}]
Table[SekiP2[n], {n, 0, 12}]
%  // Numerator   // Print
%% // Denominator // Print

{-1, 1, 1, 3, 1, 25, 1, 427, 1, 12465, 5, 555731, 691}
{1, 2, 6, 56, 30, 992, 42, 16256, 30, 261632, 66, 4192256, 2730}

Plot[{SekiP[s], SekiA[s], SekiP2[s], Seki[s]},{s, 0, 8}, 
WorkingPrecision -> 90,
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.85}], {0.9, 0.85}}]]

Formula 122

The signed Seki function.

SekiS[s_] := Exp[I Pi s / 2] (Exp[I Pi s] PolyLog[1 - s, -I] + PolyLog[1 - s, I]) s / (2^s - 4^s);
SekiS[0] := 1;
Clear[s]; SekiS[s] // FullSimplify // TeXForm

$$\frac{e^{\frac{i \pi s}{2}} s \left(e^{i \pi s} \text{Li}_{1-s}(-i)+\text{Li}_{1-s}(i)\right)}{2^s-4^s}$$

ReImPlot[SekiS[s], {s,0,6}, PlotRange -> {-1, 1.012},
PlotLegends -> Placed["Expressions", {Scaled[{0.9, 0.85}], {0.9, 0.85}}]]

See A193472 and A193473, apart from the signs.

Table[SekiS[n], {n, 0, 12}]
%  // Numerator   // Print
%% // Denominator // Print

{1, 1, -1, 3, -1, 25, -1, 427, -1, 12465, -5, 555731, -691}
{1, 2, 6, 56, 30, 992, 42, 16256, 30, 261632, 66, 4192256, 2730}

Formula 122a

The scaled signed Seki numbers.

Table[SekiS[n] / n!, {n, 0, 12}] // TeXForm

$$\left\{1,\frac{1}{2},-\frac{1}{12},\frac{1}{112},-\frac{1}{720},\frac{5}{23808},-\frac{1}{30240},\frac{61}{11704320},-\frac{1}{1209600},\frac{277}{2109800448},-\frac{1}{47900160},\frac{50521}{15212858572800},-\frac{691}{1307674368000}\right\}$$

Table[FullSimplify[SekiS[n] / n!], {n, 0, 8}]
%  // Numerator   // Print
%% // Denominator // Print

{1, 1, -1, 1, -1, 5, -1, 61, -1}
{1, 2, 12, 112, 720, 23808, 30240, 11704320, 1209600}

ez[n_] := SeriesCoefficient[Sec[t] + Tan[t], {t, 0, n}];
a[n_] := (-1)^(n-1)  If[n == 0, -1, ez[n - 1] / (4^n - 2^n)];
Table[a[n], {n, 0, 8}]

The Swiss-knife polynomials

Formula 123

Swiss knife polynomials.

Skp[-1, x_] := 1;   
Skp[n_, x_] := Sum[Binomial[n, k] Es[k] x^(n-k), {k, 0, n}];

Clear[n, x]; Skp[n, x] // TeXForm

$$\sum _{k=0}^n \binom{n}{k} x^{n-k} \text{If}\left[k+1=0,\frac{\pi }{2},2^k \text{Eg}\left(k,\frac{1}{2}\right)\right]$$

SKP := Table[Expand[FullSimplify[Skp[n, x]]], {n, 0, 7}] 
SKP // MatrixForm

Observe the sinusoidal character of the normalized SKP polynomials.

NSKP[n_, x_] := Skp[n, x] / n!;

K0 := NSKP[0,x]; K1 := NSKP[1,x]; K2 := NSKP[2,x];
K3 := NSKP[3,x]; K4 := NSKP[4,x]; K5 := NSKP[5,x];
K6 := NSKP[6,x]; K7 := NSKP[47x]; 
Plot[{K1,K2,K3,K4,K5,K6,K7}, {x, -2.5, 2.5}, PlotRange -> {-0.75, 0.75},
PlotLegends -> "Expressions"]
(* Export["Fig20SwissKnifePolynomials.eps", %] *)

Formula 124

Worpitzky representation of the SK polynomials.

a[k_] := List[1, 1, 1, 0, -1, -1, -1, 0][[Mod[k, 8] + 1]];
Table[a[n], {n, 0, 12}]

{1, 1, 1, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1}

Skpoly[-1, x_] := 1;
Skpoly[n_, x_] := Sum[a[k]*2^(-Floor[k/2]) 
                  Sum[(-1)^v Binomial[k, v] (x + v + 1)^n, 
                  {v, 0, k}], {k, 0, n}]

Table[Expand[FullSimplify[Skpoly[n, x]]], {n, 0, 7}] // MatrixForm

Formula 125, 126

Recurrence of the SK polynomials.

Clear[n, x, K]
c := {1}
P := 1
Print[P]
For[n = 1, n < 8, n++,
    c = Table[(c[[k+1]] n) / (n - 2 k), {k, 0, Floor[(n-1)/2]}];
    If[EvenQ[n], AppendTo[c, -Sum[c[[k+1]], {k, 0, Floor[(n-1)/2]}]]];
    P = Sum[c[[k+1]] x^(n - 2 k), {k, 0, Floor[n/2]}];
    Print[P]
]

1
x
      2
-1 + x
        3
-3 x + x
       2    4
5 - 6 x  + x
           3    5
25 x - 10 x  + x
          2       4    6
-61 + 75 x  - 15 x  + x
              3       5    7
-427 x + 175 x  - 21 x  + x

Asymptotics for the Bernoulli function

Formula 127

R[K_, s_] := Exp[1/2 + Sum[BernoulliB[n + 1]/((n + 1) n s^n), {n, 1, K}]];
R5[s_] := Exp[1/2 + s^(-1) / 12 - s^(-3) / 360 + s^(-5) / 1260];
R[5, s]

Formula 128

For even positive integer n:

Basyn[s_] := 4 Pi (s / (2 Pi E))^(s + 1/2) R[5, s];
Clear[n]; Basyn[n] // FullSimplify

Table[Basyn[n], {n, 20, 30, 2}] // N
Table[Abs[BernoulliB[n]], {n, 20, 30, 2}] // N

                                      6            7            8
{529.124, 6192.12, 86580.2, 1.42552 10 , 2.72982 10 , 6.01581 10 }

                                      6            7            8
{529.124, 6192.12, 86580.3, 1.42552 10 , 2.72982 10 , 6.01581 10 }

Formula 129

For positive real s:

Basy[K_, s_] := 4 Pi (s/(2 Pi Exp[1]))^(s + 1/2) R[K, s] (-Cos[s Pi / 2]);
Clear[s]; Basy[5, s] // FullSimplify (* for K=5 *)

Basy5[s_] := - 2 (Exp[(2 - 7 (s^2 - 30 s^4 + 360 s^6))/(2520 s^5)] 
              (2 Pi)^(1/2 - s) s^(1/2 + s) Cos[(Pi s) / 2]);
Clear[s]; Basy5[s]

Table[Basy5[n], {n, 20, 30, 2}] // N
Table[BernoulliB[n], {n, 20, 30, 2}] // N

                                        6             7            8
{-529.124, 6192.12, -86580.2, 1.42552 10 , -2.72982 10 , 6.01581 10 }

                                        6             7            8
{-529.124, 6192.12, -86580.3, 1.42552 10 , -2.72982 10 , 6.01581 10 }

Plot[{Basy[5, s], B[s]}, {s, 1, 13},
PlotLegends -> Placed["Expressions", {Scaled[{0.2, 0.1}], {0.2, 0.1}}],
WorkingPrecision -> 30, PlotTheme -> {"Thick", "DashedLines"}]

SekiAsy[K_, s_] := 4 Pi (s/(2 Pi Exp[1]))^(s + 1/2) R[K, s];
Clear[s]; SekiAsy[5, s] // FullSimplify

SekiAsy5[s_] := (2^(3/2 - s) Pi^(1/2 - s) s^(1/2 + s) 
                Exp[(2 - 7 (s^2 - 30 s^4 + 360 s^6))/(2520 s^5)]);
Clear[s]; SekiAsy5[s] // TeXForm

$$2^{\frac{3}{2}-s} e^{\frac{2-7 \left(360 s^6-30 s^4+s^2\right)}{2520 s^5}} \pi ^{\frac{1}{2}-s} s^{s+\frac{1}{2}}$$

atext := MaTeX["\\text{S}_{\\text{asy5}}(s)"];
btext := MaTeX["\\text{S}(s)"];

Plot[{SekiP[s], SekiAsy5[s]}, {s, 1, 13},
PlotLegends -> Placed[{btext, atext}, {0.55, 0.85}],
WorkingPrecision -> 30, PlotTheme -> {"Thick", "DashedLines"}]
(* Export["Fig33SekiApprox.eps", %] *)

Formula 130

Asymptotic expansion of the Euler function.

Easy[K_, s_] := -4 ((2 s) / (Pi Exp[1]))^(s + 1/2) R[K, s]*(-Cos[s Pi / 2]);
TeXForm[Easy[K, s]]

$$2^{s+\frac{5}{2}} \pi ^{-s-\frac{1}{2}} s^{s+\frac{1}{2}} \cos \left(\frac{\pi s}{2}\right) e^{\sum _{n=1}^K \frac{B_{n+1} s^{-n}}{n (n+1)}-s}$$

atext := MaTeX["E_{\\sigma \\text{ asy5}}(s)"];
btext := MaTeX["E_{\\sigma}(s)"];
Easy5[s_] := Easy[5, s]; 

Plot[{Esec[s], Easy5[s]}, {s, 0, 6},
WorkingPrecision -> 30, PlotRange -> {-10, 10}, 
PlotLegends -> Placed[{btext, atext}, {0.25, 0.2}],
PlotTheme -> {"Thick", "DashedLines"}]
(* Export["Fig34EulerApprox.eps", %] *)

Formula 131

Asymptotic expansion of the logarithm of the Andre function.

LogAasy[s_] := Log[4] + (1/2 + s) Log[2 s / Pi] + 
                ((2/7) - s^2 + 30 s^4 - 360 s^6) / (360 s^5);
LogAasy[s] // TeXForm

$$\frac{-360 s^6+30 s^4-s^2+\frac{2}{7}}{360 s^5}+\left(s+\frac{1}{2}\right) \log \left(\frac{2 s}{\pi }\right)+\log (4)$$

Exp[LogAasy[s]]

Plot[{Log[A[s]], LogAasy[s]}, {s, 1/2, 4}, WorkingPrecision -> 30,
PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.25, 0.85}], {0.25, 0.85}}]]

Plot[{A[s], Exp[LogAasy[s]]}, {s, 1/2, 4}, WorkingPrecision -> 30,
PlotTheme -> {"Thick", "DashedLines"},
PlotLegends -> Placed["Expressions", {Scaled[{0.25, 0.85}], {0.25, 0.85}}]]

Plot[{LogAasy[s] / Log[A[s]], 1}, {s, 9/2, 11}, 
PlotRange -> {0.99994, 1.00012}, WorkingPrecision -> 40, 
PlotLegends -> Placed["Expressions", {Scaled[{0.95, 0.85}], {0.95, 0.85}}]]
(* Export["FigXXAndreLogApprox.eps", %] *)

Formula 132

Todd = Series[z / (1 - Exp[-z]), {z, 0, 6} ]

Applications of the Swiss-knife polynomials

A122045, Euler secant numbers

Table[(-1)^n FullSimplify[Skp[n, 0]], {n, 0, 12}]

{1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765}

A155585, 2^n*E(n, 1)

Table[FullSimplify[Skp[n, 1]], {n, 0, 12}]

{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792, 0}

A163982, -2^n*(E(n, 1/2) + E(n, 1))

Table[FullSimplify[-Skp[n, 0]  - Skp[n, 1]], {n, 0, 12}]

{-2, -1, 1, 2, -5, -16, 61, 272, -1385, -7936, 50521, 353792, -2702765}

A163747, 2^n*(E(n, 1/2) - E(n, 1)).

Table[FullSimplify[Skp[n, 0]   - Skp[n, 1]], {n, 0, 12}]

{0, -1, -1, 2, 5, -16, -61, 272, 1385, -7936, -50521, 353792, 2702765}

A304980, (2^n - 4^n) B[n] / n + E[n]

Table[FullSimplify[Skp[n, 0] - Skp[n-1, 1]], {n, 1, 12}] 
Table[(2^n - 4^n) BernoulliB[n] / n + EulerE[n], {n, 1, 12}]

{-1, -2, 0, 7, 0, -77, 0, 1657, 0, -58457, 0, 3056557}

{1, -2, 0, 7, 0, -77, 0, 1657, 0, -58457, 0, 3056557}

Euler zeta numbers, A099612/A099617, 2^n*|E(n, 1/2) - E(n,1)| / n!

Table[Abs[FullSimplify[Skp[n, Mod[n, 2]]] / n!], {n, 0, 8}]
% // Numerator // Print
%% // Denominator // Print
Table[2^n Abs[EulerE[n, 1/2] - EulerE[n, 1]]/ n!, {n, 0, 8}]

{1, 1, 1, 1, 5, 2, 61, 17, 277}
{1, 1, 2, 3, 24, 15, 720, 315, 8064}

Andre numbers, A000111

Table[Abs[FullSimplify[Skp[n, Mod[n, 2]]]], {n, 0, 10}]

{1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521}

Extended Euler numbers

Table[FullSimplify[Skp[n, Mod[n, 2]]], {n, 0, 10}]

{1, 1, -1, -2, 5, 16, -61, -272, 1385, 7936, -50521}

Euler secant, A028296

Table[FullSimplify[Skp[2n, 0]], {n, 0, 8}]

{1, -1, 5, -61, 1385, -50521, 2702765, -199360981, 19391512145}

Euler tangent, A000182

Table[FullSimplify[Skp[2n + 1, 1]], {n, 0, 8}]

{1, -2, 16, -272, 7936, -353792, 22368256, -1903757312, 209865342976}

A336898 / A336899

alpha[n_] := If[n == 0, 1, n / (4^n - 2^n)]

Table[alpha[n], {n, 0, 10}]  
%  // Numerator   // Print
%% // Denominator // Print

{1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 5}
{1, 2, 6, 56, 60, 992, 672, 16256, 8160, 261632, 523776}

Bernoulli, A164555/A027642

Table[FullSimplify[Skp[n - 1, 1] alpha[n]], {n, 0, 10}]
% // Numerator // Print
%% // Denominator // Print

{1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 5}
{1, 2, 6, 1, 30, 1, 42, 1, 30, 1, 66}

Bernoulli tangent, n even, A000367/A002445

Table[FullSimplify[Skp[2n - 1, 1] alpha[2n]], {n, 0, 8}]
%  // Numerator // Print
%% // Denominator // Print

{1, 1, -1, 1, -1, 5, -691, 7, -3617}
{1, 6, 30, 42, 30, 66, 2730, 6, 510}

Bernoulli secant, n odd, A160143/A193476

Table[FullSimplify[Skp[2n, 0] alpha[2n + 1]], {n, 0, 7}]
%  // Numerator // Print
%% // Denominator // Print

{1, -3, 25, -427, 12465, -555731, 35135945, -2990414715}
{2, 56, 992, 16256, 261632, 4192256, 67100672, 1073709056}

Bernoulli extended, A193472/A193473

Table[FullSimplify[Skp[n-1, Mod[n-1, 2]] alpha[n]], {n, 0, 8}]
%  // Numerator // Print
%% // Denominator // Print

{1, 1, 1, -3, -1, 25, 1, -427, -1}
{1, 2, 6, 56, 30, 992, 42, 16256, 30}

Genocchi, A226158

Table[FullSimplify[-Skp[n-1, 1] n / 2^(n-1)], {n, 0, 12}]

{0, -1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073}

Springer, A188458

Table[FullSimplify[Skp[n, 1/2] 2^n], {n, 0, 8}]

{1, 1, -3, -11, 57, 361, -2763, -24611, 250737}

A001586 A212435

Table[FullSimplify[Skp[n, -1/2] 2^n], {n, 0, 8}]

{1, -1, -3, 11, 57, -361, -2763, 24611, 250737}

F I N