A Few Finite Trigonometric Sums by Chandan Datta and Pankaj Agrawal

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums with products of trigonometric functions can get complicated and may not have a simple expressions in a number of cases. Some of these sums have interesting properties and can have amazingly simple value. However, only some of them are available in literature. We obtain a number of such sums using method of residues.

The entire paper is available here.

Generalized Gould-Hopper Polynomials by Emil Horozov

Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and Bessel have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems with this property. In a recent paper [22] we extended the class of polynomial systems having both the property to be eigenfunctions of a differential operator (now without restriction on the order) and the 3-term recurrence relation, responsible for the orthogonality is relaxed to a d+2-term relation with \(d \in \mathbb{N}\). We found such systems also when instead of a differential operator we consider a difference one. Our polynomial systems came with many properties. In the present paper we continue to study a class of these systems in the spirit of the classical orthogonal polynomials. Their most important properties, are hypergeometric representations that we find for them. From the hypergeometric representations we derive generating functions and in some cases we find Mehler-Heine type formulas.

The paper is available here.

Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions by M. A. Shpot, R. B. Paris

We consider two integrals over \(x \in [0,1]\) involving products of the function \(\zeta_{1}(a,x) \equiv \zeta(a,x) − x^{−a}\), where \(\zeta(a,x)\) is the Hurwitz zeta function, given by
\begin{equation}
\int\limits_{0}^{1} \zeta_{1}(a,x) \zeta_{1}(b,x) \mathrm{d}x
\end{equation}
and
\begin{equation}
\int\limits_{0}^{1} \zeta_{1}(a,x) \zeta_{1}(b,1-x) \mathrm{d}x
\end{equation}

when \(\Re (a,b) \gt 1\). These integrals have been investigated recently in [23]; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function \(\zeta(x)\) and two free parameters a and b. The limiting forms of these sums when \(a+b\) takes on integer values are considered.

The paper is available here.

Josef Meixner: his life and his polynomials by Paul L. Butzer, Tom H. Koornwinder

This paper starts with a biographical sketch of the life of Josef Meixner. Then his motivations to work on orthogonal polynomials and special functions are reviewed. Meixner’s 1934 paper introducing the Meixner and Meixner-Pollaczek polynomials is discussed in detail. Truksa’s forgotten 1931 paper, which already contains the Meixner polynomials, is mentioned. The paper ends with a survey of the reception of Meixner’s 1934 paper.

The entire paper is available here.