Some infinite sums arising from the Weierstrass product theorem

Abstract

A very simple approach using series manipulations and the Weierstrass Representation Theorem yield surprisingly nice and non-trivial series identities involving special functions. In our paper we present a large number of examples of these identities. At the end, several number theoretical sums are deduced.

Introduction

There are a large number of infinite product identities connected to special functions. Our aim is to show that, employing these representations, a couple of series identities can be deduced with a simple approach. In these series the Digamma, Trigamma and Hurwitz zeta functions appear.

The first part of the paper deals with classical special functions like the above mentioned ones. In the second part, we define a new function via its infinite product expression to win new identities to the prime zeta function.

Now we introduce the main ingredients of the paper.

The nth generalized harmonic number of order r is defined as

$$H_{n\text{,}r}=\frac{1}{1^r}+\frac{1}{2^r}+\cdots +\frac{1}{n^r}(n\in \mathbb{N}≔\{1\text{,}2\text{,}3\text{,}\dots \})\text{.}$$

If n tends to infinity and $r>1$, then $H_{n\text{,}r}\to \zeta (r)$, where $\zeta$ is the Riemann zeta function. Its extension is the Hurwitz zeta function (or generalized zeta function) and it is defined as

$$\zeta (s\text{,}a)={\displaystyle \sum _{n=0}^{\infty }}\frac{1}{(n+a{)}^s}(s>1\text{;}a\in \mathbb{R}⧹{\mathbb{Z}}_{-})\text{,}$$

where $\mathbb{R}$ and ${\mathbb{Z}}_{-}$ denote the set of real numbers and nonpositive integers, respectively.

The Psi or Digamma function $\psi (x)$ is defined by an infinite series [6]:

$$\psi (x)=-\gamma +{\displaystyle \sum _{n=0}^{\infty }}\left(\frac{1}{n+1}-\frac{1}{n+x}\right)(x\in \mathbb{R}⧹{\mathbb{Z}}_{-})\text{,}$$

where $\gamma =0.5772\dots$ is the Euler–Mascheroni constant [3], [6]. Moreover, the Trigamma function ${\psi }^{\prime }$ is the derivative of the Digamma function. Its series representation comes from (1):

$${\psi }^{\prime }(x)={\displaystyle \sum _{n=0}^{\infty }}\frac{1}{(n+x{)}^2}=\zeta (2\text{,}x)(x\in \mathbb{R}⧹{\mathbb{Z}}_{-})\text{.}$$