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# Generalized Glaisher-Kinkelin constants and Ramanujan summation of series

Marc-Antoine Coppo

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Marc-Antoine Coppo. Generalized Glaisher-Kinkelin constants and Ramanujan summation of series. Research in Number Theory, 2024, 10, pp.8. 10.1007/s40993-023-00505-2. http://dx.doi.org/10.1007/s40993-023-00505-2

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# Generalized Glaisher-Kinkelin constants and Ramanujan summation of series

Marc-Antoine Coppo

<figure><img src="image_2.png" /></figure>

## To cite this version:

Marc-Antoine Coppo. Generalized Glaisher-Kinkelin constants and Ramanujan summation of series. Research in Number Theory, 2024, 10, pp.8. 10.1007/s40993-023-00505-2. hal-03197403v23

HAL Id: hal-03197403

https://univ-cotedazur.hal.science/hal-03197403v23

Submitted on 3 Jun 2024

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
# Generalized Glaisher-Kinkelin constants and Ramanujan summation of series

Marc-Antoine Coppo*

Université Côte d'Azur, CNRS, LJAD (UMR 7351), Nice, France

*Research in Number Theory, 10 (2024)*

**Abstract** We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind.

**Keywords** Bendersky-Adamchik constants; Bendersky's generalized gamma functions; Hurwitz zeta function; Bernoulli numbers of the second kind; Ramanujan summation of series.

## 1 Introduction: from Stirling to Bendersky and beyond

The story of the constants known today as the Bendersky-Adamchik constants begins with the famous Stirling formula for the factorial:

$$n! \sim \sqrt{2\pi} n^{n+\frac{1}{2}} e^{-n} \quad \text{as} \quad n \to \infty$$

which dates back from the middle of the 18th century. In fact, Stirling never explicitly stated this formula. The first appearance of this result occurs in a letter from Euler to Goldbach dated June 1744. The constant $\sqrt{2\pi}$ is commonly referred to as the Stirling constant.

<footnote>*Corresponding author. Email address: coppo@unice.fr</footnote>
# Generalized Glaisher-Kinkelin constants and Ramanujan summation of series

Marc-Antoine Coppo*

Université Côte d'Azur, CNRS, LJAD (UMR 7351), Nice, France

*Research in Number Theory, 10 (2024)*

**Abstract** We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind.

**Keywords** Bendersky-Adamchik constants; Bendersky's generalized gamma functions; Hurwitz zeta function; Bernoulli numbers of the second kind; Ramanujan summation of series.

## 1 Introduction: from Stirling to Bendersky and beyond

The story of the constants known today as the Bendersky-Adamchik constants begins with the famous Stirling formula for the factorial:

$$n! \sim \sqrt{2\pi} n^{n+\frac{1}{2}} e^{-n} \quad \text{as} \quad n \to \infty$$

which dates back from the middle of the 18th century. In fact, Stirling never explicitly stated this formula. The first appearance of this result occurs in a letter from Euler to Goldbach dated June 1744. The constant $\sqrt{2\pi}$ is commonly referred to as the Stirling constant.

<footnote>*Corresponding author. Email address: coppo@unice.fr</footnote>
A similar but less well-known formula, due to Glaisher, also applies to the hyperfactorial:

$$ \prod_{\nu=1}^{n} \nu^{\nu} = 1^{1} 2^{2} \cdots n^{n} \sim A n^{\frac{n^2}{2} + \frac{n}{2} + \frac{1}{12}} e^{-\frac{n^2}{4}} \quad \text{as} \quad n \to \infty, $$

where the constant $A = 1.282427...$ is the Glaisher-Kinkelin constant [9, Prop. 11].

In an important article published in 1933, Bendersky [2] studied the product $\prod_{\nu=1}^{n} \nu^{\nu^k}$ for $k = 0, 1, 2, ...$ which reduces to the classical factorial when $k = 0$, and to the hyperfactorial when $k = 1$. For his purpose, Bendersky introduced a natural generalization $\Gamma_k$ of the classical $\Gamma$-function, defined on the positive real axis, whose fundamental properties are $\Gamma_k(1) = 1$ and

$$ \Gamma_k(x + 1) = x^{x^k} \Gamma_k(x) \quad \text{for all } x > 0. $$

In particular, for each positive integer $n$,

$$ \Gamma_k(n + 1) = \prod_{\nu=1}^{n} \nu^{\nu^k} \quad \text{for} \quad k = 0, 1, 2, \dots $$

Note that $\Gamma_0 = \Gamma$, and $\Gamma_1$ is the Kinkelin hyperfactorial $K$-function [11].

Bendersky showed, for any integer $k \ge 0$, the existence of a constant $A_k$ and a couple of polynomials $P_k$ and $Q_k$ of degree $k + 1$ such that

$$ \Gamma_k(x + 1) \sim A_k x^{P_k(x)} e^{-Q_k(x)} \quad \text{as} \quad x \to +\infty. $$

Using the Euler-Maclaurin summation formula, Bendersky managed to evaluate the constants $A_k$ and the polynomials $P_k$ and $Q_k$ for $k \le 4$.

Unaware of Bendersky's work and following an idea of Milnor, Kurokawa and Ochiai [10, Thm. 2] rediscovered much more later the function $\Gamma_k$ and expressed appropriately this function in terms of the derivative of the Hurwitz zeta function $\zeta(s, x)$ at $s = -k$. More precisely, they established the following representation<fnref n="1" />

$$ \ln \Gamma_k(x) = \zeta'(-k, x) - \zeta'(-k) \quad \text{for } x > 0 \text{ and } k \ge 0, $$

which extends a classical representation due to Lerch in the case $k = 0$ [6, p. 78].

Our initial interest in these constants comes from their interesting interpretation in terms of the Ramanujan summation of certain divergent series, as explained

<footnote n="1">1. According to Kellner [9, Rem. 27], this expression of $\ln \Gamma_k$ is due to Alexeiewsky in the special case where $x = n + 1$ is an integer.</footnote>
A similar but less well-known formula, due to Glaisher, also applies to the hyperfactorial:

$$ \prod_{\nu=1}^{n} \nu^{\nu} = 1^{1} 2^{2} \cdots n^{n} \sim A n^{\frac{n^2}{2} + \frac{n}{2} + \frac{1}{12}} e^{-\frac{n^2}{4}} \quad \text{as} \quad n \to \infty, $$

where the constant $A = 1.282427...$ is the Glaisher-Kinkelin constant [9, Prop. 11].

In an important article published in 1933, Bendersky [2] studied the product $\prod_{\nu=1}^{n} \nu^{\nu^k}$ for $k = 0, 1, 2, ...$ which reduces to the classical factorial when $k = 0$, and to the hyperfactorial when $k = 1$. For his purpose, Bendersky introduced a natural generalization $\Gamma_k$ of the classical $\Gamma$-function, defined on the positive real axis, whose fundamental properties are $\Gamma_k(1) = 1$ and

$$ \Gamma_k(x + 1) = x^{x^k} \Gamma_k(x) \quad \text{for all } x > 0. $$

In particular, for each positive integer $n$,

$$ \Gamma_k(n + 1) = \prod_{\nu=1}^{n} \nu^{\nu^k} \quad \text{for} \quad k = 0, 1, 2, \dots $$

Note that $\Gamma_0 = \Gamma$, and $\Gamma_1$ is the Kinkelin hyperfactorial $K$-function [11].

Bendersky showed, for any integer $k \ge 0$, the existence of a constant $A_k$ and a couple of polynomials $P_k$ and $Q_k$ of degree $k + 1$ such that

$$ \Gamma_k(x + 1) \sim A_k x^{P_k(x)} e^{-Q_k(x)} \quad \text{as} \quad x \to +\infty. $$

Using the Euler-Maclaurin summation formula, Bendersky managed to evaluate the constants $A_k$ and the polynomials $P_k$ and $Q_k$ for $k \le 4$.

Unaware of Bendersky's work and following an idea of Milnor, Kurokawa and Ochiai [10, Thm. 2] rediscovered much more later the function $\Gamma_k$ and expressed appropriately this function in terms of the derivative of the Hurwitz zeta function $\zeta(s, x)$ at $s = -k$. More precisely, they established the following representation<fnref n="1" />

$$ \ln \Gamma_k(x) = \zeta'(-k, x) - \zeta'(-k) \quad \text{for } x > 0 \text{ and } k \ge 0, $$

which extends a classical representation due to Lerch in the case $k = 0$ [6, p. 78].

Our initial interest in these constants comes from their interesting interpretation in terms of the Ramanujan summation of certain divergent series, as explained

<footnote n="1">1. According to Kellner [9, Rem. 27], this expression of $\ln \Gamma_k$ is due to Alexeiewsky in the special case where $x = n + 1$ is an integer.</footnote>
in Section 4. Thanks to the properties of the function ln $Γ_k$, we are in a position to completely elucidate the link between the logarithm of the constant $A_k$ and the sum of the divergent series $\sum_{n \ge 1} n^k \ln n$ in the sense of Ramanujan's summation method, following the exposition in [4]. More precisely, using the notations of [4], we show (see Theorem 1) that for all integers $k \ge 0$,

$$ \ln A_k = \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n + \frac{H_k B_{k+1}}{k+1} + \frac{1}{(k+1)^2}, $$

where, in this expression, $H_k$ and $B_k$ denote respectively the kth harmonic number and the kth Bernoulli number.

## 2 Bendersky-Adamchik constants

**Definition 1.** The sequence of real numbers {$A_k$}$_{k \ge 0}$ can be defined as follows [13, Eq. (1.1)-(1.6)]: for all integers $k \ge 0$,

$$ \ln A_k := \lim_{n \to \infty} \left\{ \sum_{\nu=1}^{n} \nu^k \ln \nu - P_k(n) \ln n + Q_k(n) \right\}, \quad (1) $$

where $(P_k, Q_k)$ is a couple of polynomials of degree $k+1$ with rational coefficients such that

$$ (P_0, Q_0) = (x + \frac{1}{2}, x), $$

$$ (P_1, Q_1) = (\frac{x^2}{2} + \frac{x}{2} + \frac{1}{12}, \frac{x^2}{4}), $$

and whose general expression is given for $k \ge 2$ by

$$ P_k(x) = \frac{x^{k+1}}{k+1} + \frac{x^k}{2} + \sum_{r=1}^{\left[\frac{k+1}{2}\right]} \frac{B_{2r}}{(2r)!} \left( \prod_{j=1}^{2r-1} (k-j+1) \right) x^{k+1-2r}, $$

and

$$ Q_k(x) = \frac{x^{k+1}}{(k+1)^2} - \sum_{r=1}^{\left[\frac{k+1}{2}\right]+\frac{(-1)^{k-1}}{2}} \frac{B_{2r}}{(2r)!} \left\{ \prod_{j=1}^{2r-1} (k-j+1) \sum_{j=1}^{2r-1} \frac{1}{k-j+1} \right\} x^{k+1-2r}, $$

where {$B_{2r}$}$_{r \ge 0}$ is the sequence of (even) Bernoulli numbers. The numbers $A_k$ (for $k = 0, 1, 2, ...$) are called the *Bendersky-Adamchik constants* because Bendersky introduced these constants for the first time in 1933 [2], and Adamchik [1]
in Section 4. Thanks to the properties of the function ln $\Gamma_k$, we are in a position to completely elucidate the link between the logarithm of the constant $A_k$ and the sum of the divergent series $\sum_{n \ge 1} n^k \ln n$ in the sense of Ramanujan's summation method, following the exposition in [4]. More precisely, using the notations of [4], we show (see Theorem 1) that for all integers $k \ge 0$,

$$ \ln A_k = \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n + \frac{H_k B_{k+1}}{k+1} + \frac{1}{(k+1)^2}, $$

where, in this expression, $H_k$ and $B_k$ denote respectively the kth harmonic number and the kth Bernoulli number.

## 2 Bendersky-Adamchik constants

**Definition 1.** The sequence of real numbers {$A_k$}$_{k \ge 0}$ can be defined as follows [13, Eq. (1.1)-(1.6)]: for all integers $k \ge 0$,

$$ \ln A_k := \lim_{n \to \infty} \left\{ \sum_{\nu=1}^{n} \nu^k \ln \nu - P_k(n) \ln n + Q_k(n) \right\}, \quad (1) $$

where $(P_k, Q_k)$ is a couple of polynomials of degree $k+1$ with rational coefficients such that

$$ (P_0, Q_0) = (x + \frac{1}{2}, x), $$

$$ (P_1, Q_1) = (\frac{x^2}{2} + \frac{x}{2} + \frac{1}{12}, \frac{x^2}{4}), $$

and whose general expression is given for $k \ge 2$ by

$$ P_k(x) = \frac{x^{k+1}}{k+1} + \frac{x^k}{2} + \sum_{r=1}^{\left\lfloor \frac{k+1}{2} \right\rfloor} \frac{B_{2r}}{(2r)!} \left( \prod_{j=1}^{2r-1} (k-j+1) \right) x^{k+1-2r}, $$

and

$$ Q_k(x) = \frac{x^{k+1}}{(k+1)^2} - \sum_{r=1}^{\left\lfloor \frac{k+1}{2} \right\rfloor + \frac{(-1)^{k-1}}{2}} \frac{B_{2r}}{(2r)!} \left\{ \prod_{j=1}^{2r-1} (k-j+1) \sum_{j=1}^{2r-1} \frac{1}{k-j+1} \right\} x^{k+1-2r}, $$

where {$B_{2r}$}$_{r \ge 0}$ is the sequence of (even) Bernoulli numbers. The numbers $A_k$ (for $k = 0, 1, 2, ...$) are called the *Bendersky-Adamchik constants* because Bendersky introduced these constants for the first time in 1933 [2], and Adamchik [1]
rediscovered them more than 60 years later, giving a nice expression of $\ln A_k$ in terms of the derivatives of the Riemann zeta function [1, Prop. 4]. More precisely, this expression (called Adamchik's formula in the remainder of this article) is the following:

$$ \ln A_k = \frac{H_k B_{k+1}}{k+1} - \zeta'(-k), \qquad (2) $$

where $H_k$ and $B_k$ denote respectively the $k$th harmonic number (with the usual convention $H_0 = 0$) and the $k$th Bernoulli number. In particular, $A_0$ is the Stirling constant, and $A_1 = A$ is the Glaisher-Kinkelin constant [1, 9, 12, 13].

### 3 Adamchik's formula revisited

The following identities:

$$ \zeta'(-2k) = (-1)^k \frac{(2k)!}{2(2\pi)^{2k}} \zeta(2k+1) \quad (k \ge 1), $$

and

$$ \zeta'(1-2k) = (-1)^{k+1} \frac{(2k)!}{k(2\pi)^{2k}} \zeta'(2k) + \frac{B_{2k}}{2k} (H_{2k-1} - \gamma - \ln 2\pi) \quad (k \ge 1), $$

where $\gamma = 0.577215...$ is Euler's constant, are easily derived by differentiation of the functional equation of the zeta function [12, p. 384]. Furthermore, combined with the famous Euler formula [6, p. 17]

$$ \zeta(2k) = (-1)^{k+1} \frac{(2\pi)^{2k}}{2(2k)!} B_{2k} \quad (k \ge 1), $$

these identities allow us to rephrase formula (2) as follows:

$$ \ln A_{2k} = \frac{B_{2k}}{4} \left( \frac{\zeta(2k+1)}{\zeta(2k)} \right) \quad (k \ge 1), \qquad (3) $$

and

$$ \ln A_{2k-1} = \frac{B_{2k}}{2k} \left( \gamma + \ln 2\pi - \frac{\zeta'(2k)}{\zeta(2k)} \right) \quad (k \ge 1). \qquad (4) $$

Thus, Adamchik's formula means that $\ln A_k$ is closely related to $\zeta(k+1)/\zeta(k)$ when $k$ is positive and even, and to $\zeta'(k+1)/\zeta(k+1)$ in the odd case. For example,

$$ \ln A = \frac{1}{12} \left( \gamma + \ln 2\pi - \frac{\zeta'(2)}{\zeta(2)} \right) \quad \text{and} \quad \ln A_2 = \frac{1}{24} \left( \frac{\zeta(3)}{\zeta(2)} \right). $$
rediscovered them more than 60 years later, giving a nice expression of $\ln A_k$ in terms of the derivatives of the Riemann zeta function [1, Prop. 4]. More precisely, this expression (called Adamchik's formula in the remainder of this article) is the following:

$$ \ln A_k = \frac{H_k B_{k+1}}{k+1} - \zeta'(-k), \qquad (2) $$

where $H_k$ and $B_k$ denote respectively the $k$th harmonic number (with the usual convention $H_0 = 0$) and the $k$th Bernoulli number. In particular, $A_0$ is the Stirling constant, and $A_1 = A$ is the Glaisher-Kinkelin constant [1, 9, 12, 13].

### 3 Adamchik's formula revisited

The following identities:

$$ \zeta'(-2k) = (-1)^k \frac{(2k)!}{2(2\pi)^{2k}} \zeta(2k+1) \quad (k \ge 1), $$

and

$$ \zeta'(1 - 2k) = (-1)^{k+1} \frac{(2k)!}{k(2\pi)^{2k}} \zeta'(2k) + \frac{B_{2k}}{2k} (H_{2k-1} - \gamma - \ln 2\pi) \quad (k \ge 1), $$

where $\gamma = 0.577215...$ is Euler's constant, are easily derived by differentiation of the functional equation of the zeta function [12, p. 384]. Furthermore, combined with the famous Euler formula [6, p. 17]

$$ \zeta(2k) = (-1)^{k+1} \frac{(2\pi)^{2k}}{2(2k)!} B_{2k} \quad (k \ge 1), $$

these identities allow us to rephrase formula (2) as follows:

$$ \ln A_{2k} = \frac{B_{2k}}{4} \left( \frac{\zeta(2k+1)}{\zeta(2k)} \right) \quad (k \ge 1), \qquad (3) $$

and

$$ \ln A_{2k-1} = \frac{B_{2k}}{2k} \left( \gamma + \ln 2\pi - \frac{\zeta'(2k)}{\zeta(2k)} \right) \quad (k \ge 1). \qquad (4) $$

Thus, Adamchik's formula means that $\ln A_k$ is closely related to $\zeta(k+1)/\zeta(k)$ when $k$ is positive and even, and to $\zeta'(k+1)/\zeta(k+1)$ in the odd case. For example,

$$ \ln A = \frac{1}{12} \left( \gamma + \ln 2\pi - \frac{\zeta'(2)}{\zeta(2)} \right) \quad \text{and} \quad \ln A_2 = \frac{1}{24} \left( \frac{\zeta(3)}{\zeta(2)} \right). $$
## 4 Link with the Ramanujan summation of series

Formula (1) strongly suggests that the constant $\ln A_k$ is closely related to the $\mathcal{R}$-sum of the divergent series $\sum_{n \ge 1} n^k \ln n$ (i.e. the sum of the series in the sense of Ramanujan's summation method, following the notations and exposition in [4]). More precisely, we have the following nice result:

**Theorem 1.** For all integers $k \ge 0$,

$$ \ln A_k = \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n + \frac{H_k B_{k+1}}{k+1} + \frac{1}{(k+1)^2}. \qquad (5) $$

*Proof.* For any real number $x$ with $x > -1$ and integer $k \ge 0$, let us consider the function

$$ \varphi_k(x) := \ln \Gamma_k(x+1) = \zeta'(-k, x+1) - \zeta'(-k). $$

The function $\varphi_k$ satisfies both $\varphi_k(0) = 0$ and the difference equation

$$ \varphi_k(x) - \varphi_k(x-1) = x^k \ln x \quad (x > 0). $$

Thus, from [4, Eq. (1.30)], we can write the identity

$$ \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n = \int_0^1 \varphi_k(x) dx = \int_0^1 \ln \Gamma_k(x+1) dx. $$

We then make use of the following identity [2, Eq. (V'k), p. 280]:

$$ \int_0^1 \ln \Gamma_k(x+1) dx = \ln A_k - \frac{H_k B_{k+1}}{k+1} - \frac{1}{(k+1)^2} $$

which allows us to deduce formula (5).

<figure><img src="image_3.png" /></figure>

We also derive from (5) and Adamchik's formula (2) the following elegant corollary:

**Corollary 1.** For any $k \ge 0$,

$$ \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n = -\zeta'(-k) - \frac{1}{(k+1)^2}. \qquad (6) $$
## 4 Link with the Ramanujan summation of series

Formula (1) strongly suggests that the constant $\ln A_k$ is closely related to the $\mathcal{R}$-sum of the divergent series $\sum_{n \ge 1} n^k \ln n$ (i.e. the sum of the series in the sense of Ramanujan's summation method, following the notations and exposition in [4]). More precisely, we have the following nice result:

**Theorem 1.** For all integers $k \ge 0$,

$$ \ln A_k = \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n + \frac{H_k B_{k+1}}{k+1} + \frac{1}{(k+1)^2}. \qquad (5) $$

*Proof.* For any real number $x$ with $x > -1$ and integer $k \ge 0$, let us consider the function

$$ \varphi_k(x) := \ln \Gamma_k(x+1) = \zeta'(-k, x+1) - \zeta'(-k). $$

The function $\varphi_k$ satisfies both $\varphi_k(0) = 0$ and the difference equation

$$ \varphi_k(x) - \varphi_k(x-1) = x^k \ln x \quad (x > 0). $$

Thus, from [4, Eq. (1.30)], we can write the identity

$$ \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n = \int_0^1 \varphi_k(x) dx = \int_0^1 \ln \Gamma_k(x+1) dx. $$

We then make use of the following identity [2, Eq. (V'k), p. 280]:

$$ \int_0^1 \ln \Gamma_k(x+1) dx = \ln A_k - \frac{H_k B_{k+1}}{k+1} - \frac{1}{(k+1)^2} $$

which allows us to deduce formula (5).

<figure><img src="image_4.png" /></figure>

We also derive from (5) and Adamchik's formula (2) the following elegant corollary:

**Corollary 1.** For any $k \ge 0$,

$$ \sum_{n \ge 1}^{\mathcal{R}} n^k \ln n = -\zeta'(-k) - \frac{1}{(k+1)^2}. \qquad (6) $$
**Example 1.**

$$ \sum_{n \ge 1}^{\mathcal{R}} \ln n = \frac{1}{2} \ln 2\pi - 1 = -\zeta'(0) - 1, $$

$$ \sum_{n \ge 1}^{\mathcal{R}} n \ln n = \ln A - \frac{1}{3} = \frac{1}{12} \left( \gamma + \ln 2\pi - \frac{\zeta'(2)}{\zeta(2)} - 4 \right), $$

$$ \sum_{n \ge 1}^{\mathcal{R}} n^2 \ln n = \ln A_2 - \frac{1}{9} = \frac{1}{24} \left( \frac{\zeta(3)}{\zeta(2)} \right) - \frac{1}{9}. $$

*Remark 1.* The following formulas can also be shown (cf. [4, p. 82]):

$$ \sum_{n \ge 1}^{\mathcal{R}} H_n = \frac{3}{2}\gamma - \frac{1}{2}\ln 2\pi + \frac{1}{2}, $$

$$ \sum_{n \ge 1}^{\mathcal{R}} nH_n = \frac{5}{12}\gamma - \frac{1}{2}\ln 2\pi + \ln A + \frac{19}{24}, $$

$$ \sum_{n \ge 1}^{\mathcal{R}} n^2 H_n = \frac{1}{3}\gamma - \frac{1}{2}\ln 2\pi + 2\ln A - \ln A_2 + \frac{13}{24}, $$

and more generally, for any integer $k \ge 2$,

$$ \sum_{n \ge 1}^{\mathcal{R}} n^k H_n = \left( \frac{1 - B_{k+1}}{k+1} \right) \gamma + \sum_{j=0}^{k} (-1)^{j+1} \binom{k}{j} \ln A_j + \frac{1}{k} + c_k, $$

with

$$ c_k = \begin{cases} (k-1) \frac{B_k}{2k} & \text{if } k \text{ is even,} \\ \sum_{j=0}^{k} \binom{k}{j} \frac{B_j B_{k+1-j}}{(k+1-j)^2} & \text{if } k \text{ is odd.} \end{cases} $$

**5 New expansion of $\ln A_k$ in convergent series**

In his seminal work, Bendersky [2, pp. 295–299] presented two different (cumber-
some) expansions of the logarithm $L_k = \ln A_k$ in convergent series. In this section,
we give a new one, of a completely different kind, involving a convergent series
with only rational terms.

For convenience, we first introduce the sequence of positive rational numbers
$\{\lambda_n\}_{n \ge 1}$ (called non-alternating Cauchy numbers in [5]) defined by

$$ \lambda_n := \left| \sum_{k=1}^{n} \frac{s(n, k)}{k+1} \right| \quad (n \ge 1), $$
**Example 1.**

$$ \sum_{n \ge 1}^{\mathcal{R}} \ln n = \frac{1}{2} \ln 2\pi - 1 = -\zeta'(0) - 1, $$

$$ \sum_{n \ge 1}^{\mathcal{R}} n \ln n = \ln A - \frac{1}{3} = \frac{1}{12} \left( \gamma + \ln 2\pi - \frac{\zeta'(2)}{\zeta(2)} - 4 \right), $$

$$ \sum_{n \ge 1}^{\mathcal{R}} n^2 \ln n = \ln A_2 - \frac{1}{9} = \frac{1}{24} \left( \frac{\zeta(3)}{\zeta(2)} \right) - \frac{1}{9}. $$

*Remark 1.* The following formulas can also be shown (cf. [4, p. 82]):

$$ \sum_{n \ge 1}^{\mathcal{R}} H_n = \frac{3}{2}\gamma - \frac{1}{2}\ln 2\pi + \frac{1}{2}, $$

$$ \sum_{n \ge 1}^{\mathcal{R}} n H_n = \frac{5}{12}\gamma - \frac{1}{2}\ln 2\pi + \ln A + \frac{19}{24}, $$

$$ \sum_{n \ge 1}^{\mathcal{R}} n^2 H_n = \frac{1}{3}\gamma - \frac{1}{2}\ln 2\pi + 2\ln A - \ln A_2 + \frac{13}{24}, $$

and more generally, for any integer $k \ge 2$,

$$ \sum_{n \ge 1}^{\mathcal{R}} n^k H_n = \left( \frac{1 - B_{k+1}}{k+1} \right) \gamma + \sum_{j=0}^{k} (-1)^{j+1} \binom{k}{j} \ln A_j + \frac{1}{k} + c_k, $$

with

$$ c_k = \begin{cases} (k-1) \frac{B_k}{2k} & \text{if } k \text{ is even,} \\ \sum_{j=0}^{k} \binom{k}{j} \frac{B_j B_{k+1-j}}{(k+1-j)^2} & \text{if } k \text{ is odd.} \end{cases} $$

**5 New expansion of $\ln A_k$ in convergent series**

In his seminal work, Bendersky [2, pp. 295–299] presented two different (cumber-
some) expansions of the logarithm $L_k = \ln A_k$ in convergent series. In this section,
we give a new one, of a completely different kind, involving a convergent series
with only rational terms.

For convenience, we first introduce the sequence of positive rational numbers
$\{\lambda_n\}_{n \ge 1}$ (called non-alternating Cauchy numbers in [5]) defined by

$$ \lambda_n := \left| \sum_{k=1}^{n} \frac{s(n, k)}{k+1} \right| \quad (n \ge 1), $$
where $s(n, k)$ denotes the (signed) Stirling numbers of the first kind. The first ones are the following:

$$ \lambda_1 = \frac{1}{2}, \lambda_2 = \frac{1}{6}, \lambda_3 = \frac{1}{4}, \lambda_4 = \frac{19}{30}, \lambda_5 = \frac{9}{4}, \lambda_6 = \frac{863}{84}, \text{ etc.} $$

The numbers $\lambda_n$ are closely related to the Bernoulli numbers of the second kind $b_n$ [8], which can be defined by the generating function

$$ \frac{x}{\ln(x+1)} = \sum_{n=0}^{\infty} b_n x^n, $$

through the simple relation

$$ \lambda_n = (-1)^{n-1}n!b_n = n!|b_n| \quad (n \ge 1). $$

The following identity is already known (see e.g. [8, Prop. 2]):

$$ \ln A_0 = \frac{1}{2} \ln(2\pi) = \sum_{n=2}^{\infty} \frac{\lambda_n}{n!(n-1)} + \frac{1}{2}\gamma + \frac{1}{2}. $$

Combining Adamchik's formula with a result previously given in [8] enables us to considerably expand the scope of this formula through the following theorem:

**Theorem 2.** For all integers $k$ and $r$ with $0 \le r \le k$, let $S(k, r)$ be the Stirling numbers of the second kind

$$ S(k, r) = \frac{1}{r!} \sum_{j=0}^{r} (-1)^{r-j} \binom{r}{j} j^k, $$

and $\sigma_r$ be the shifted Mascheroni series

$$ \sigma_r := \sum_{n=r+1}^{\infty} \frac{\lambda_n}{n!(n-r)} \quad (r \ge 0). $$

Then, for all integers $k \ge 1$, we have

$$ \ln A_k = (-1)^k \sum_{r=1}^{k} (-1)^r r! S(k, r) \sigma_{r+1} + \frac{B_{k+1}}{k+1} (H_{k+1} + \gamma). \quad (7) $$

In particular,

$$ \ln A = \sigma_2 + \frac{1}{12}\gamma + \frac{1}{8}. $$

*Proof.* Formula (7) results from the decomposition of $\zeta'(-k)$ given by [8, Prop. 3] and from Adamchik's formula (2). $\square$
where $s(n, k)$ denotes the (signed) Stirling numbers of the first kind. The first ones are the following:

$$ \lambda_1 = \frac{1}{2}, \lambda_2 = \frac{1}{6}, \lambda_3 = \frac{1}{4}, \lambda_4 = \frac{19}{30}, \lambda_5 = \frac{9}{4}, \lambda_6 = \frac{863}{84}, \text{ etc.} $$

The numbers $\lambda_n$ are closely related to the Bernoulli numbers of the second kind $b_n$ [8], which can be defined by the generating function

$$ \frac{x}{\ln(x+1)} = \sum_{n=0}^{\infty} b_n x^n, $$

through the simple relation

$$ \lambda_n = (-1)^{n-1}n!b_n = n!|b_n| \quad (n \ge 1). $$

The following identity is already known (see e.g. [8, Prop. 2]):

$$ \ln A_0 = \frac{1}{2} \ln(2\pi) = \sum_{n=2}^{\infty} \frac{\lambda_n}{n!(n-1)} + \frac{1}{2}\gamma + \frac{1}{2}. $$

Combining Adamchik's formula with a result previously given in [8] enables us to considerably expand the scope of this formula through the following theorem:

**Theorem 2.** For all integers $k$ and $r$ with $0 \le r \le k$, let $S(k, r)$ be the Stirling numbers of the second kind

$$ S(k, r) = \frac{1}{r!} \sum_{j=0}^{r} (-1)^{r-j} \binom{r}{j} j^k, $$

and $\sigma_r$ be the shifted Mascheroni series

$$ \sigma_r := \sum_{n=r+1}^{\infty} \frac{\lambda_n}{n!(n-r)} \quad (r \ge 0). $$

Then, for all integers $k \ge 1$, we have

$$ \ln A_k = (-1)^k \sum_{r=1}^{k} (-1)^r r! S(k, r) \sigma_{r+1} + \frac{B_{k+1}}{k+1} (H_{k+1} + \gamma). \quad (7) $$

In particular,

$$ \ln A = \sigma_2 + \frac{1}{12}\gamma + \frac{1}{8}. $$

*Proof.* Formula (7) results from the decomposition of $\zeta'(-k)$ given by [8, Prop. 3] and from Adamchik's formula (2). $\square$
**Corollary 2.** For any positive integer $k$, we have

$$ \ln A_{2k} = \sum_{n=2k+2}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k} \frac{(-1)^r r! S(2k, r)}{n-1-r} \right\} + C_{2k}, \quad (8) $$

and

$$ \ln A_{2k-1} = \sum_{n=2k+1}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k-1} \frac{(-1)^{r-1} r! S(2k-1, r)}{n-1-r} \right\} + \frac{B_{2k}}{2k} (H_{2k} + \gamma) + C_{2k-1}, \quad (9) $$

where the constants $C_k$ are given by $C_1 = 0$, and

$$ C_k = (-1)^k \sum_{r=1}^{k-1} (-1)^r r! S(k, r) \sum_{j=r+2}^{k+1} \frac{\lambda_j}{j!(j-1-r)} \quad (k \ge 2). $$

**Example 2.**

$$ \ln A_2 = \sum_{n=4}^{\infty} \frac{\lambda_n (n-1)}{n! (n-2)(n-3)} - \frac{1}{24}, $$

$$ \ln A_3 = \sum_{n=5}^{\infty} \frac{\lambda_n n(n-1)}{n! (n-2)(n-3)(n-4)} - \frac{1}{120}\gamma - \frac{29}{240}, $$

$$ \ln A_4 = \sum_{n=6}^{\infty} \frac{\lambda_n (n-1)^2 (n+4)}{n! (n-2)(n-3)(n-4)(n-5)} - \frac{113}{480}. $$

The relation between $\ln A_k$ and the $\mathcal{R}$-sum $\sum_{n \ge 1} n^k \ln n$ provided by Theorem 1 allows us to derive from Corollary 2 above the corresponding formulas:

**Corollary 3.** For all integers $k \ge 1$,

$$ \sum_{n \ge 1} n^{2k} \ln n = \sum_{n=2k+2}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k} \frac{(-1)^r r! S(2k, r)}{n-1-r} \right\} - \frac{1}{(2k+1)^2} + C_{2k}, \quad (10) $$

and

$$ \sum_{n \ge 1} n^{2k-1} \ln n = \sum_{n=2k+1}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k-1} \frac{(-1)^{r-1} r! S(2k-1, r)}{n-1-r} \right\} + \frac{B_{2k}}{2k}\gamma + D_{2k}, \quad (11) $$

with

$$ D_{2k} = \frac{B_{2k} - 1}{(2k)^2} + C_{2k-1} \quad (k \ge 1). $$
**Corollary 2.** For any positive integer $k$, we have

$$ \ln A_{2k} = \sum_{n=2k+2}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k} \frac{(-1)^r r! S(2k, r)}{n-1-r} \right\} + C_{2k}, \quad (8) $$

and

$$ \ln A_{2k-1} = \sum_{n=2k+1}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k-1} \frac{(-1)^{r-1} r! S(2k-1, r)}{n-1-r} \right\} + \frac{B_{2k}}{2k} (H_{2k} + \gamma) + C_{2k-1}, \quad (9) $$

where the constants $C_k$ are given by $C_1 = 0$, and

$$ C_k = (-1)^k \sum_{r=1}^{k-1} (-1)^r r! S(k, r) \sum_{j=r+2}^{k+1} \frac{\lambda_j}{j!(j-1-r)} \quad (k \ge 2). $$

**Example 2.**

$$ \ln A_2 = \sum_{n=4}^{\infty} \frac{\lambda_n (n-1)}{n! (n-2)(n-3)} - \frac{1}{24}, $$

$$ \ln A_3 = \sum_{n=5}^{\infty} \frac{\lambda_n n(n-1)}{n! (n-2)(n-3)(n-4)} - \frac{1}{120}\gamma - \frac{29}{240}, $$

$$ \ln A_4 = \sum_{n=6}^{\infty} \frac{\lambda_n (n-1)^2 (n+4)}{n! (n-2)(n-3)(n-4)(n-5)} - \frac{113}{480}. $$

The relation between $\ln A_k$ and the $\mathcal{R}$-sum $\sum_{n \ge 1} n^k \ln n$ provided by Theorem 1 allows us to derive from Corollary 2 above the corresponding formulas:

**Corollary 3.** For all integers $k \ge 1$,

$$ \sum_{n \ge 1} n^{2k} \ln n = \sum_{n=2k+2}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k} \frac{(-1)^r r! S(2k, r)}{n-1-r} \right\} - \frac{1}{(2k+1)^2} + C_{2k}, \quad (10) $$

and

$$ \sum_{n \ge 1} n^{2k-1} \ln n = \sum_{n=2k+1}^{\infty} \frac{\lambda_n}{n!} \left\{ \sum_{r=1}^{2k-1} \frac{(-1)^{r-1} r! S(2k-1, r)}{n-1-r} \right\} + \frac{B_{2k}}{2k}\gamma + D_{2k}, \quad (11) $$

with

$$ D_{2k} = \frac{B_{2k} - 1}{(2k)^2} + C_{2k-1} \quad (k \ge 1). $$
# 6 Bendersky-Adamchik constants and Blagouchine's integral

In this additional section, we complete an unpublished short note of Blagouchine [3] by establishing a link with the Bendersky-Adamchik constants through a binomial sum.

**Theorem 3.** For any integer $k \ge 0$, let $\mathcal{J}_k$ be the complex integral ("Blagouchine's integral")

$$ \mathcal{J}_k := \int_{-\infty}^{+\infty} \frac{\zeta(\frac{1}{2} + ix)}{(2k + 1 + 2ix) \cosh(\pi x)} dx. $$

Then we have $\mathcal{J}_0 = -1$, and for all $k \ge 1$,

$$ \mathcal{J}_k = \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \ln A_j - \frac{1}{k} - \frac{1}{(k+1)^2}. \qquad (12) $$

*Proof.* Theorem 3 results directly from the following two lemmas. $\square$

**Lemma 1.** For any integer $k \ge 0$, let $\nu_k$ be the infinite alternating series

$$ \nu_k := \sum_{n=2}^{\infty} (-1)^n \frac{\zeta(n)}{n+k}. $$

For all $k \ge 0$, we have the relation:

$$ \nu_k = \frac{\gamma}{k+1} - \frac{1}{(k+1)^2} - \mathcal{J}_k. \qquad (13) $$

In particular, since $\nu_0 = \gamma$ (a classical formula due to Euler), we have $\mathcal{J}_0 = -1$.

*Proof.* Formula (13) is a special case of [3, Thm. 1] in the case where $\omega = k$ is an integer. We give a new direct proof. For $k \ge 0$, let us consider the function

$$ f_k(z) = \frac{\zeta(z)}{(k+z) \sin(\pi z)}. $$

This function $f_k$ has poles at integers $n \in \mathbb{Z}$. For $n \ge 2$, the residue of $f_k$ at $z = n$ is

$$ \mathrm{Res}(f_k; n) = \frac{(-1)^n \zeta(n)}{(n+k)\pi}. $$

For $n = 1$, $f_k$ has a double pole and

$$ \mathrm{Res}(f_k; 1) = -\frac{1}{\pi} \left( \frac{\gamma}{k+1} - \frac{1}{(k+1)^2} \right). $$
# 6 Bendersky-Adamchik constants and Blagouchine's integral

In this additional section, we complete an unpublished short note of Blagouchine [3] by establishing a link with the Bendersky-Adamchik constants through a binomial sum.

**Theorem 3.** For any integer $k \ge 0$, let $J_k$ be the complex integral ("Blagouchine's integral")

$$J_k := \int_{-\infty}^{+\infty} \frac{\zeta(\frac{1}{2} + ix)}{(2k + 1 + 2ix) \cosh(\pi x)} dx.$$

Then we have $J_0 = -1$, and for all $k \ge 1$,

$$J_k = \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \ln A_j - \frac{1}{k} - \frac{1}{(k+1)^2} \quad (12)$$

*Proof.* Theorem 3 results directly from the following two lemmas. $\square$

**Lemma 1.** For any integer $k \ge 0$, let $\nu_k$ be the infinite alternating series

$$\nu_k := \sum_{n=2}^{\infty} (-1)^n \frac{\zeta(n)}{n+k}.$$

For all $k \ge 0$, we have the relation:

$$\nu_k = \frac{\gamma}{k+1} - \frac{1}{(k+1)^2} - J_k. \quad (13)$$

In particular, since $\nu_0 = \gamma$ (a classical formula due to Euler), we have $J_0 = -1$.

*Proof.* Formula (13) is a special case of [3, Thm. 1] in the case where $\omega = k$ is an integer. We give a new direct proof. For $k \ge 0$, let us consider the function

$$f_k(z) = \frac{\zeta(z)}{(k+z)\sin(\pi z)}.$$

This function $f_k$ has poles at integers $n \in \mathbb{Z}$. For $n \ge 2$, the residue of $f_k$ at $z = n$ is

$$\mathrm{Res}(f_k; n) = \frac{(-1)^n \zeta(n)}{(n+k)\pi}.$$

For $n = 1$, $f_k$ has a double pole and

$$\mathrm{Res}(f_k; 1) = -\frac{1}{\pi} \left( \frac{\gamma}{k+1} - \frac{1}{(k+1)^2} \right).$$
If $q$ is a positive odd integer with $1 < q$, then, by the residue theorem, we have

$$ \int_{\mathrm{Re}(z)=1/2} f_k(z) dz - \int_{\mathrm{Re}(z)=q/2} f_k(z) dz = -2i\pi \sum_{\frac{1}{2} < n < \frac{q}{2}} \mathrm{Res}(f_k; n). \quad (*) $$

Moreover, there is a positive constant $C$ such that

$$ \left| \int_{\mathrm{Re}(z)=q/2} f_k(z) dz \right| \le C \int_{-\infty}^{+\infty} \frac{1}{\left(\left(k + \frac{q}{2}\right)^2 + t^2\right)^{\frac{1}{2}} (e^{\pi t} + e^{-\pi t})} dt, $$

and thus, by the dominated convergence theorem, we have

$$ \left| \int_{\mathrm{Re}(z)=q/2} f_k(z) dz \right| \to 0 \quad \text{as} \quad q \to +\infty. $$

Therefore, taking the limit in ($*$), we obtain

$$ \int_{\mathrm{Re}(z)=1/2} f_k(z) dz = -2i\pi \sum_{n>\frac{1}{2}} \mathrm{Res}(f_k; n). $$

This last identity allows us to deduce formula (13). $\square$

**Lemma 2.** For all $k \ge 1$, we have the identity

$$ \nu_k = \frac{\gamma}{k+1} + \frac{1}{k} - \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \ln A_j. \quad (14) $$

*Proof.* From [7, Prop. 1], we know that

$$ \nu_k = \frac{\gamma}{k+1} + \frac{1}{k} + \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \zeta'(-j) + \sum_{j=0}^{k-1} \binom{k}{j} \frac{H_j B_{j+1}}{j+1} \quad (k \ge 1). $$

Rephrasing this later expression thanks to Adamchik's formula (2) leads to the equivalent formula (14). $\square$

**Remark 2.** Since $\mathcal{J}_0 = -1$, we can also rewrite (12) in a slightly different manner:

$$ \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \ln A_j = \mathcal{J}_k - \frac{k^2 + 3k + 1}{k(k+1)^2} \mathcal{J}_0 \quad (k \ge 1). \quad (15) $$
If $q$ is a positive odd integer with $1 < q$, then, by the residue theorem, we have

$$ \int_{\text{Re}(z)=1/2} f_k(z) dz - \int_{\text{Re}(z)=q/2} f_k(z) dz = -2i\pi \sum_{\frac{1}{2} < n < \frac{q}{2}} \text{Res}(f_k; n). \quad (*) $$

Moreover, there is a positive constant $C$ such that

$$ \left| \int_{\text{Re}(z)=q/2} f_k(z) dz \right| \le C \int_{-\infty}^{+\infty} \frac{1}{\left(\left(k + \frac{q}{2}\right)^2 + t^2\right)^{\frac{1}{2}} (e^{\pi t} + e^{-\pi t})} dt, $$

and thus, by the dominated convergence theorem, we have

$$ \left| \int_{\text{Re}(z)=q/2} f_k(z) dz \right| \to 0 \quad \text{as} \quad q \to +\infty. $$

Therefore, taking the limit in ($*$), we obtain

$$ \int_{\text{Re}(z)=1/2} f_k(z) dz = -2i\pi \sum_{n>\frac{1}{2}} \text{Res}(f_k; n). $$

This last identity allows us to deduce formula (13). $\square$

**Lemma 2.** For all $k \ge 1$, we have the identity

$$ \nu_k = \frac{\gamma}{k+1} + \frac{1}{k} - \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \ln A_j. \quad (14) $$

*Proof.* From [7, Prop. 1], we know that

$$ \nu_k = \frac{\gamma}{k+1} + \frac{1}{k} + \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \zeta'(-j) + \sum_{j=0}^{k-1} \binom{k}{j} \frac{H_j B_{j+1}}{j+1} \quad (k \ge 1). $$

Rephrasing this later expression thanks to Adamchik's formula (2) leads to the equivalent formula (14). $\square$

**Remark 2.** Since $J_0 = -1$, we can also rewrite (12) in a slightly different manner:

$$ \sum_{j=0}^{k-1} (-1)^j \binom{k}{j} \ln A_j = \mathcal{J}_k - \frac{k^2 + 3k + 1}{k(k+1)^2} \mathcal{J}_0 \quad (k \ge 1). \quad (15) $$
**Example 3.** For small positive values of $k$, we derive from (12) the following identities:

$$J_1 = \frac{1}{2} \ln 2\pi - \frac{5}{4},$$

$$J_2 = \frac{1}{3} \ln 2\pi - \frac{1}{6}\gamma - \frac{11}{18} + \frac{1}{6}\frac{\zeta'(2)}{\zeta(2)},$$

$$J_3 = \frac{1}{4} \ln 2\pi - \frac{1}{4}\gamma - \frac{19}{48} + \frac{1}{4}\frac{\zeta'(2)}{\zeta(2)} + \frac{1}{8}\frac{\zeta(3)}{\zeta(2)},$$

and, for $k \ge 3$, using (3) and (4), we get the general formula:

$$J_k = \frac{1}{k+1} \ln 2\pi - \frac{(k-1)}{2(k+1)} \gamma - \frac{k^2 + 3k + 1}{k(k+1)^2} \\ + \sum_{j=1}^{\left\lfloor \frac{k}{2} \right\rfloor} \binom{k}{2j-1} \frac{B_{2j} \zeta'(2j)}{2j \zeta(2j)} + \frac{1}{4} \sum_{j=1}^{\left\lfloor \frac{k-1}{2} \right\rfloor} \binom{k}{2j} \frac{B_{2j} \zeta(2j+1)}{\zeta(2j)}. \quad (16)$$

## References

[1] V. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math. 100 (1998), 191–199.

[2] L. Bendersky, Sur la fonction gamma généralisée, Acta Math. 61 (1933), 263–322.

[3] I. V. Blagouchine, A complement to a recent paper on some infinite sums with the zeta values, preprint, 2020. Available at https://arxiv.org/abs/2001.00108

[4] B. Candelpergher, Ramanujan Summation of Divergent Series, Lecture Notes in Math. 2185, Springer, 2017.

[5] B. Candelpergher and M-A. Coppo, A new class of identities involving Cauchy numbers, harmonic numbers and zeta values, Ramanujan J. 27 (2012), 305–328.

[6] H. Cohen, Number Theory, Volume II: Analytic and Modern Tools, Graduate Texts in Math., vol. 240, Springer, 2007.

[7] M-A. Coppo, A note on some alternating series involving zeta and multiple zeta values, J. Math. Anal. Appl. 475 (2019), 1831–1841.

[8] M-A. Coppo and P. T. Young, On shifted Mascheroni series and hyperharmonic numbers, J. Number Theory, 169 (2016), 1–20.
**Example 3.** For small positive values of $k$, we derive from (12) the following identities:

$$J_1 = \frac{1}{2} \ln 2\pi - \frac{5}{4},$$

$$J_2 = \frac{1}{3} \ln 2\pi - \frac{1}{6}\gamma - \frac{11}{18} + \frac{1}{6}\frac{\zeta'(2)}{\zeta(2)},$$

$$J_3 = \frac{1}{4} \ln 2\pi - \frac{1}{4}\gamma - \frac{19}{48} + \frac{1}{4}\frac{\zeta'(2)}{\zeta(2)} + \frac{1}{8}\frac{\zeta(3)}{\zeta(2)},$$

and, for $k \ge 3$, using (3) and (4), we get the general formula:

$$J_k = \frac{1}{k+1} \ln 2\pi - \frac{(k-1)}{2(k+1)} \gamma - \frac{k^2 + 3k + 1}{k(k+1)^2} \\ + \sum_{j=1}^{\left\lfloor \frac{k}{2} \right\rfloor} \binom{k}{2j-1} \frac{B_{2j} \zeta'(2j)}{2j \zeta(2j)} + \frac{1}{4} \sum_{j=1}^{\left\lfloor \frac{k-1}{2} \right\rfloor} \binom{k}{2j} \frac{B_{2j} \zeta(2j+1)}{\zeta(2j)}. \quad (16)$$

## References

[1] V. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math. 100 (1998), 191–199.

[2] L. Bendersky, Sur la fonction gamma généralisée, Acta Math. 61 (1933), 263–322.

[3] I. V. Blagouchine, A complement to a recent paper on some infinite sums with the zeta values, preprint, 2020. Available at https://arxiv.org/abs/2001.00108

[4] B. Candelpergher, Ramanujan Summation of Divergent Series, Lecture Notes in Math. 2185, Springer, 2017.

[5] B. Candelpergher and M-A. Coppo, A new class of identities involving Cauchy numbers, harmonic numbers and zeta values, Ramanujan J. 27 (2012), 305–328.

[6] H. Cohen, Number Theory, Volume II: Analytic and Modern Tools, Graduate Texts in Math., vol. 240, Springer, 2007.

[7] M-A. Coppo, A note on some alternating series involving zeta and multiple zeta values, J. Math. Anal. Appl. 475 (2019), 1831–1841.

[8] M-A. Coppo and P. T. Young, On shifted Mascheroni series and hyperharmonic numbers, J. Number Theory, 169 (2016), 1–20.
[9] B. C. Kellner, On asymptotic constants related to products of Bernoulli numbers, *Integers* **9**, (2009), 83–106.

[10] N. Kurokawa and H. Ochiai, Generalized Kinkelin's formulas *Kodai Math. J.* **30** (2007), 195–212.

[11] J. Sondow and P. Hadjicostas, The generalized-Euler-constant function $\gamma(z)$ and a generalization of Somos's quadratic recurrence constant, *J. Math. Anal. Appl.* **332** (2007), 292–314.

[12] M. Perkins and R. A. Van Gorder, Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series, *Ramanujan J.* **49** (2019), 371–389.

[13] W. Wang, Some asymptotic expansions of hyperfactorial functions and generalized Glaisher-Kinkelin constants, *Ramanujan J.* **43** (2017), 513–533.
[9] B. C. Kellner, On asymptotic constants related to products of Bernoulli numbers, *Integers* **9**, (2009), 83–106.

[10] N. Kurokawa and H. Ochiai, Generalized Kinkelin's formulas *Kodai Math. J.* **30** (2007), 195–212.

[11] J. Sondow and P. Hadjicostas, The generalized-Euler-constant function $\gamma(z)$ and a generalization of Somos's quadratic recurrence constant, *J. Math. Anal. Appl.* **332** (2007), 292–314.

[12] M. Perkins and R. A. Van Gorder, Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series, *Ramanujan J.* **49** (2019), 371–389.

[13] W. Wang, Some asymptotic expansions of hyperfactorial functions and generalized Glaisher-Kinkelin constants, *Ramanujan J.* **43** (2017), 513–533.