A note on some constants related to the zeta-function and their relationship with the Gregory coefficients

Abstract

In this article, new series for the first and second Stieltjes constants (also known as generalized Euler’s constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind and the Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler’s constant $\gamma$ and the constant $\ln 2\pi$ are given, and yet another generalization of Euler’s constant is proposed and various formulas for the calculation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation.

Appendix: Yet another generalization of Euler’s constant

The numbers $\kappa _p:=\sum |G_n|\,n^{-p-1}$, where the summation extends over positive integers n, may also be regarded as one of the possible generalizations of Euler’s constant (since $\kappa _0=\gamma _0=\gamma$ and $\kappa _{-1}=\gamma _{-1}=1$).¹³$^{,}$¹⁴ These constants, which do not seem to be reducible to “classical mathematical constants”, admit several interesting representations as stated in the following proposition.

Proposition 1

Generalized Euler’s constants $\kappa _{p}:=\sum |G_n|\,n^{-p-1}$, where the summation extends over positive integers n, admit the following representations:

$$\begin{aligned} \kappa _{p}\displaystyle= & {} \,\frac{(-1)^{p}}{{\varGamma }(p+1)}\int \limits _0^1 \left\{ \frac{1}{\ln (1-x)}+ \frac{1}{x} \right\} \ln ^{p} x\; \mathrm{d}x, \quad \mathrm{Re}\,p > -1 \end{aligned}$$

(29)

$$\begin{aligned} \displaystyle= & {} \underbrace{\int \limits _0^1\cdots \int \limits _0^1 }_{p\text {-fold}} \left\{ {{\mathrm{li}}}\left( 1-\prod _{k=1}^{p}x_k \right) +\gamma + \sum _{k=1}^{p}\ln x_k \right\} \, \frac{\mathrm{d}x_1 \cdots \mathrm{d}x_{p}}{x_1 \cdots x_{p}}, \quad p = 1, 2, 3,\ldots \end{aligned}$$

(30)

$$\begin{aligned} \displaystyle= & {} \sum _{k=2}^{\infty } \frac{(-1)^{k}}{k} \sum _{n=p+1}^{\infty } \frac{\big |S_1(n,p+1)\big |}{n!\,n^{k-1}}, \quad p = 0, 1,2,\ldots \end{aligned}$$

(31)

$$\begin{aligned} \displaystyle= & {} \sum _{k=2}^{\infty } \frac{(-1)^{k}}{k} \sum _{n=p}^{\infty }\, \frac{P_{p}(H_{n}^{(1)}, -H_{n}^{(2)},\dots , (-1)^{p-1}H_{n}^{(p)}) }{(n+1)^{k}}, \quad p = 0, 1, 2,\ldots \end{aligned}$$

(32)

$$\begin{aligned} \displaystyle= & {} \sum _{n\geqslant 1}^{\mathcal {R}} \frac{1}{n} \sum _{n\geqslant n_{1}\geqslant \cdots \geqslant n_{p}\geqslant 1}\frac{1}{n_{1}\ldots n_{p}} \nonumber \\= & {} \sum _{n\geqslant 1}^{\mathcal {R}} \frac{P_{p}\big (H_{n}^{(1)}, H_{n}^{(2)},\dots ,H_{n}^{(p)}\big ) }{n},\quad p= 1, 2, 3, \ldots \qquad , \end{aligned}$$

(33)

where ${{\mathrm{li}}}$ is the integral logarithm function, $H^{(m)}_n:=\sum _{k=1}^n k^{-m}$ stands for the generalized harmonic number and $P_m$ denotes the sequence of polynomials

$$\begin{aligned} P_0= & {} 1,\quad P_1(x_1) = x_1, \quad P_2(x_1, x_2) = \tfrac{1}{2}\left( x_1^2 + x_2\right) ,\\ P_3 (x_1, x_2, x_3)= & {} \tfrac{1}{6}\left( x_1^3 + 3 x_1 x_2 + 2x_3\right) , \quad \ldots ^{15} \end{aligned}$$

¹⁵ In particular, for the series $\kappa _1$ mentioned in Theorem 1 and Remark 2, this gives

$$\begin{aligned} \kappa _1\,= & {} \sum _{n=1}^{\infty }\frac{\big |G_n\big |}{n^2} \displaystyle =\, -\int \limits _0^1 \left\{ \frac{1}{\ln (1-x)}+ \frac{1}{x} \right\} \ln x\; \mathrm{d}x \end{aligned}$$

(34)

$$\begin{aligned} \displaystyle= & {} \, \int \limits _0^1 \frac{\,-{{\mathrm{li}}}(1-x) + \gamma + \ln x\,}{x}\, \mathrm{d}x \, = \int \limits _0^\infty \Big \{ -{{\mathrm{li}}}\big (1-\mathrm{e}^{-x}\big ) + \gamma - x \Big \}\, \mathrm{d}x \end{aligned}$$

(35)

$$\begin{aligned} \displaystyle= & {} \sum _{k=2}^{\infty } \frac{(-1)^{k}}{k}\sum _{n=2}^{\infty }\frac{H_{n-1}}{n^{k}}\,= \sum _{n\geqslant 1}^{\mathcal {R}} \frac{H_n}{n}. \end{aligned}$$

(36)

Moreover, we also have

$$\begin{aligned} \displaystyle \kappa _1&\displaystyle =\gamma _1 + \frac{\gamma ^2}{2} - \frac{\pi ^2}{12} + \int \limits _0^1 \frac{{\varPsi }(x+1) + \gamma }{x}\,\mathrm{d}x \end{aligned}$$

(37)

$$\begin{aligned}&\displaystyle = \frac{\gamma ^2}{2} + \frac{\pi ^2}{12} -\frac{1}{2} + \frac{1}{2}\int \limits _0^1 {\varPsi }^2(x+1) \,\mathrm{d}x, \end{aligned}$$

(38)

where ${\varPsi }$ denotes the digamma function (logarithmic derivative of the ${\varGamma }$-function).

Proof of formula (29)

We first write the generating equation for Gregory’s coefficients, Eq. (7), in the following form

$$\begin{aligned} \frac{1}{\ln (1-x)}+ \frac{1}{x} \, =\sum _{n=1}^\infty |G_{n}|\, x^{n-1}, \qquad |x|<1. \end{aligned}$$

(39)

Multiplying both sides by $\ln ^{p} x$, integrating over the unit interval and changing the order of summation and integration¹⁶ yield

$$\begin{aligned} \int \limits _0^1 \left\{ \frac{1}{\ln (1-x)}+ \frac{1}{x} \right\} \ln ^{p} x\; \mathrm{d}x\, =\, \sum _{n=1}^\infty |G_{n}| \int \limits _0^1 x^{n-1} \ln ^{p} x \; \mathrm{d}x,\qquad \mathrm{Re}\,p>-1. \end{aligned}$$

(40)

The last integral may be evaluated as follows. Considering Legendre’s integral $\,{\varGamma }(p+1)\,=\int t^{p} \mathrm{e}^{-t} \mathrm{d}t\,$ taken over $[0,\infty )$ and making a change of variable $\,t=-(s+1)\ln x\,$, we have

$$\begin{aligned} \,\int \limits _0^1 x^s \ln ^p x \; \mathrm{d}x \,=\, (-1)^p\frac{{\varGamma }(p+1)}{\, (s+1)^{p+1}}, \quad \mathrm{Re}\,s>-1 ,\quad \mathrm{Re}\,p>-1. \end{aligned}$$

(41)

Inserting this formula into (40) and setting $n-1$ instead of s yield (29).$\square$

Proof of formula (30)

Putting in (39) $\,x=x_1x_2\cdots x_{p+1}\,$ and integrating over the volume $[0,1]^{p+1}$, where $p\in {{\mathbb {N}}}$, on the one hand, we have

$$\begin{aligned} \underbrace{\int \limits _0^1\cdots \int \limits _0^1}_{(p+1)\text {-fold}} \sum _{n=1}^{\infty } |G_n| \big (x_1x_2\cdots x_{p+1}\big )^{n-1} \mathrm{d}x_1 \cdots \mathrm{d}x_{p+1} \,= \sum _{n=1}^{\infty }\frac{|G_n|}{n^{p+1}}. \end{aligned}$$

(42)

On the other hand

$$\begin{aligned} \int \limits _0^1 \left\{ \frac{1}{\ln (1-xy)}+ \frac{1}{xy} \right\} \, \mathrm{d}x\, =\,- \frac{\,{{\mathrm{li}}}(1-y) - \gamma - \ln y\,}{y}. \end{aligned}$$

Taking instead of y the product $\,x_1x_2\cdots x_{p}\,$ and setting $\,x=x_{p+1}\,$, and then integrating p times over the unit hypercube and equating the result with (42) yield (30). $\square$

Proof of formulas (31) and (32)

Writing in the generating equation (11) $-x$ instead of z, multiplying it by $\ln ^m x/x$ and integrating over the unit interval, we obtain the following relation¹⁷

$$\begin{aligned} {\varOmega }(k,m) = (-1)^{m+k} m! \, k! \sum _{n=k}^{\infty } \frac{\big |S_1(n,k)\big |}{n!\,n^{m+1}}, \end{aligned}$$

where

$$\begin{aligned} {\varOmega }(k,m) \, := \int \limits _0^1 \frac{\ln ^k(1-x)\,\ln ^m x}{x} \, \mathrm{d}x,\qquad k,m\in \mathbb {N}. \end{aligned}$$

By integration by parts, it may be readily shown that

$$\begin{aligned} {\varOmega }(k,m) \,=\,\frac{k}{m+1}{\varOmega }(m+1,k-1), \end{aligned}$$

and thus, we deduce the duality formula:

$$\begin{aligned} \sum _{n=k}^{\infty } \frac{\big |S_1(n,k)\big |}{n!\,n^{m+1}} = \sum _{n=m+1}^{\infty } \frac{\big |S_1(n,m+1)\big |}{n!\,n^{k}}. \end{aligned}$$

Furthermore, the MacLaurin series expansion of $\mathrm{e}^z$ with $z=\ln (1-x)$ gives

$$\begin{aligned} x + \ln (1-x) = - \sum _{k=1}^{\infty }\frac{\ln ^{k+1}(1-x)}{(k+1)!},\qquad x<1. \end{aligned}$$

Hence

$$\begin{aligned} \int \limits _0^1 \left\{ \frac{1}{\ln (1-x)}+ \frac{1}{x} \right\} \ln ^{m} x\; \mathrm{d}x&= - \int \limits _0^1 \sum _{k=1}^{\infty }\frac{\ln ^{k+1}(1-x)}{(k+1)!}\cdot \frac{\ln ^m x}{\ln (1-x)}\cdot \frac{\mathrm{d}x}{x}\\&= - \sum _{k=1}^{\infty } \frac{{\varOmega }(k,m)}{(k+1)!} \, \\&= (-1)^m m! \sum _{k=1}^{\infty } \frac{(-1)^{k+1}}{(k+1)} \sum _{n=m+1}^{\infty } \frac{\big |S_1(n,m+1)\big |}{n!\,n^{k}}, \end{aligned}$$

which is identical with (31) if setting $m=p$. Furthermore, it is well known that

$$\begin{aligned} \frac{\big |S_1(n+1,m+1)\big |}{n!}\, =\, P_{m}\big (H_{n}^{(1)}, -H_{n}^{(2)},\ldots , (-1)^{m-1}H_{n}^{(m)}\big ), \end{aligned}$$

see [17, p. 217], [37, p. 1395], [31, p. 425, Eq. (43)], [4, Eq. (16)], which immediately gives (32) and completes the proof.$\square$

Proof of formula (33)

This formula straightforwardly follows from the fact that $\kappa _p=F_p(1)$, see [11, p. 307, 318 et seq.], where $F_{p}(s)$ is the special function defined by

$$\begin{aligned} F_p(s) := \sum _{n=1}^{\infty } \frac{|G_{n}|}{n^{p}} D\left( \frac{1}{\,x^s}\right) (n) = \sum _{n=0}^\infty \frac{|G_{n+1}|}{(n+1)^p}\sum _{k=0}^{n} (-1)^k \left( {\begin{array}{c}n\\ k\end{array}}\right) (k+1)^{-s}. \end{aligned}$$

$\square$

Proof of formulas (37) and (38)

These formulas immediately follow from [10, Eqs. (3.21) and (3.23)] and (36). $\square$