The aim of this paper is to provide a new class of series identities in the form of four general results. The results are established with the help of generalizatons of the classical Kummers summation theorem obtained earlier by Rakha and Rathie. Results obtained earlier by Srivastava, Bailey and Rathie et al. follow special cases of our main findings.
In this article an alternative infinite product for a special class of the entire functions are studied by using some results of the Laguerre-P{o}lya entire functions. The zeros for a class of the special even entire functions are discussed in detail. It is proved that the infinite product and series representations for the hyperbolic and trigonometric cosine functions, which are coming from Euler, are our special cases.
In this note we study the flint hill series of the form begin{align} sum limits_{n=1}^{infty}frac{1}{(sin^2n) n^3} onumber end{align}via a certain method. The method works essentially by erecting certain pillars sufficiently close to the terms in the series and evaluating the series at those spots. This allows us to relate the convergence and the divergence of the series to other series that are somewhat tractable. In particular we show that the convergence of the flint hill series relies very heavily on the condition that for any small $epsilon>0$ begin{align} bigg|sum limits_{i=0}^{frac{n+1}{2}}sum limits_{j=0}^{i}(-1)^{i-j}binom{n}{2i+1} binom{i}{j}bigg|^{2s} leq |(sin^2n)|n^{2s+2-epsilon} onumber end{align}for some $sin mathbb{N}$.
We present a number of identities involving standard and associated Laguerre polynomials. They include double-, and triple-lacunary, ordinary and exponential generating functions of certain classes of Laguerre polynomials.
Let $qge3$ be an integer, $chi$ be a Dirichlet character modulo $q$, and $L(s,chi)$ denote the Dirichlet $L$-functions corresponding to $chi$. In this paper, we show some special function series, and give some new identities for the Dirichlet $L$-functions involving Gauss sums. Specially, we give specific identities for $L(2,chi)$.