Multiple Argument Euler Sum Identities

Greenberg March 4, 2025 in News - 3 Minutes

Euler, a notable mathematician of the seventeenth century, published many brilliant results in various branches of mathematics. In his works on series, after many illustrious scientists of his time gave various approximations to the Basel problem, Euler was able to show that

$\sum_{n = 1}^{\infty} \frac{1}{n^{2}} = ζ (2) = \frac{π^{2}}{6} .$

Furthermore, Euler proved from the properties of the Bernoulli numbers, $B_{0} = 1, B_{1} = - \frac{1}{2}, B_{2} = \frac{1}{6}, B_{2 r + 1} = 0,$ for $r \geq 1,$ given by the generating function

$\frac{w}{exp (w) - 1} = \sum_{j \geq 0} \frac{w^{j}}{j!} B j, for |w| < 2 π$

and the classical Euler, Bernoulli relation (see, [1], p. 166), that

$ζ (2 r) = \frac{{(- 1)}^{r + 1} B_{2 r} 2^{2 r - 1} π^{2 r}}{(2 r)!} .$

The Riemann zeta function, $ζ (t)$ , is defined as

$ζ (t) : = \{\begin{matrix} \sum_{j \geq 1} \frac{1}{j^{t}} = \frac{1}{1 - 2^{- t}} \sum_{j \geq 1} \frac{1}{{(2 j - 1)}^{t}}, ℜ (t) > 1 \\ \frac{1}{1 - 2^{1 - t}} \sum_{j \geq 1} \frac{{(- 1)}^{j - 1}}{j^{t}}, ℜ (t) > 0, t \neq 1 \end{matrix}$

and the Dirichlet eta function, $η (t)$ , is defined by

$η (t) : = \sum_{n ⩾ 1} \frac{{(- 1)}^{n - 1}}{n^{t}} = (1 - 2^{1 - t}) ζ (t), (ℜ (t) > 0),$

where $ℜ (t)$ defines the real part of $t .$ The harmonic numbers $H_{n}^{(p)}$ of integer order p are defined as

$H_{n}^{(p)} : = \sum_{r = 1}^{n} \frac{1}{r^{p}}, p \in C and n \in N$

where $C$ and $N$ are the sets of complex numbers and positive integers, respectively.

The harmonic numbers of order one

H_{q},

are given by the following, for

q \in R^{+}

$\begin{matrix} H_{q} & = γ + ψ (q + 1) \\ = γ + ψ (q) + \frac{1}{q}, H_{0} : = 0 . \end{matrix}$

The term $γ$ represents the familiar Euler–Mascheroni constant (see, e.g., [1], Section 1.2), and $ψ (t)$ denotes the digamma (or psi) function defined by

$\begin{matrix} ψ (t) & : & = \frac{d}{d t} (log Γ (t)) = \frac{Γ^{'} (t)}{Γ (t)} (t \in C ∖ Z_{⩽ 0}), and \\ ψ (t + 1) & : & = ψ (t) + \frac{1}{t}, \end{matrix}$

where $Γ (t)$ is the familiar Gamma function (see, e.g., [1], Section 1.1). Following the notation of Flajolet and Salvy [2], we define classes of linear Euler harmonic sums of the type

$S_{p, t}^{+ +} (q) : = \sum_{n = 1}^{\infty} \frac{H_{q n}^{(p)}}{n^{t}}, S_{p, t}^{+ -} (q) : = \sum_{n = 1}^{\infty} {(- 1)}^{n + 1} \frac{H_{q n}^{(p)}}{n^{t}},$

(1)

for $q \in R^{+} ∖ 0$ and with the integer $p + t$ designated as the weight. We define

$S_{p, t}^{+ +} (1) : = S_{p, t}^{+ +}, S_{p, t}^{+ -} (1) : = S_{p, t}^{+ -} .$

In this paper, we shall obtain linear combinations of (1) in closed form in terms of special functions, such as

$\begin{matrix} S_{1, t + 1}^{+ +} (\frac{1}{q}) + S_{1, t + 1}^{+ -} (\frac{1}{q}) + \frac{2 {(- 1)}^{t}}{q^{t}} S_{1, t + 1}^{+ +} (q) - \frac{{(- 1)}^{t}}{q^{t}} S_{1, t + 1}^{+ +} (\frac{q}{2}) \\ = 2 \sum_{j = 1}^{t - 1} \frac{{(- 1)}^{j + 1}}{q^{j}} ζ (j + 1) λ (t + 1 - j), \forall t \in N, \end{matrix}$

(2)

where the Dirichlet lambda function $λ (s)$ is defined as the term-wise arithmetic mean of the Dirichlet eta function and the Riemann zeta function:

$\begin{matrix} λ (m) & = \frac{η (m) + ζ (m)}{2} = lim_{n \to \infty} O_{n}^{(m)} \\ = lim_{n \to \infty} (\sum_{j = 1}^{n} \frac{1}{{(2 j - 1)}^{m}}) \\ = \sum_{j = 1}^{\infty} \frac{1}{{(2 j - 1)}^{m}}, (ℜ (m) > 1) . \end{matrix}$

For the case where $q = 1$ , we recall the following known results

$S_{1, t}^{+ +} = (1 + \frac{t}{2}) ζ (t + 1) - \frac{1}{2} \sum_{j = 1}^{t - 2} ζ (j + 1) ζ (t - j),$

(3)

due to Euler [3]. For the alternating case, for odd weight $1 + t$ , Sitaramachandrarao [4] published

$2 S_{1, t}^{+ -} = (t + 1) η (t + 1) - ζ (t + 1) - 2 \sum_{j = 1}^{\frac{t}{2} - 1} ζ (2 j) ζ (t + 1 - 2 j),$

(4)

and recently Alzer and Choi [5] obtained the following nice result, for $p \in N ∖ 1$

$2 S_{p, 1}^{+ -} = p ζ (p + 1) - ζ (t + 1) - 2 \sum_{j = 1}^{p} η (j) η (p + 1 - j) .$

The polylogarithm function

{Li}_{p} (z)

of order

p,

and for each integer

p \geq 1,

is defined by the following (see, e.g., [1], p. 198)

${Li}_{p} (z) = \sum_{m = 1}^{\infty} \frac{z^{m}}{m^{p}} (| z | ⩽ 1; p \in N \geq 2)$

(5)

and

${Li}_{1} (z) = - log (1 - z), (| z | ⩽ 1) .$

There are many interesting and significant results associated with Euler harmonic sum identities, some of which may be seen in the works of [6,7,8,9,10,11,12,13,14]. Linear Euler harmonic sums play an important role in the evaluation of integral equations in many areas of science research such as combinatorics and statistical plasma physics, especially in the context of the Sommerfeld temperature expansion of electronic entropy; see [15,16]. The majority of the published works dealing with Euler harmonic sums of the type (6) deal with the case where $q = 1 .$