A New Generalization of the Alternating Harmonic Series

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned by linear combinations of Lerch zeta functions is characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This result is an analogue of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the $(a, c)$-variables having the Lerch zeta function as an eigenfunction.

In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz–Lerch zeta functions. Inversion formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization of digamma and gamma functions in terms of Hurwitz–Lerch zeta functions, and consider its properties. In all the sections, various results are always proved by multiplication and inversion formulas.

Series acceleration formulas are obtained for Dirichlet series with periodic coefficients. Special cases include Ramanujan's formula for the values of the Riemann zeta function at the odd positive integers exceeding two, and related formulas for values of Dirichlet L-series and the Lerch zeta function.

We explore a recently opened approach to the study of zeta functions, namely the approach of fractional calculus. By utilising the machinery of fractional derivatives and integrals, which have rarely been applied in analytic number theory before, we are able to obtain some fractional differential relations and finally a partial differential equation of fractional type which is satisfied by the Lerch zeta function.

This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step of arithmetic progression, there is a closed non-empty subset of the space of analytic functions defined in the critical strip such that its functions can be approximated by discrete shifts of the Lerch zeta function. The set of those shifts is infinite, and it has a positive density. For the proof, the weak convergence of probability measures in the space of analytic functions is applied.

We give explicit approximate functional equations for various classes of Dirichlet series, such as Lerch zeta function or Dirichlet L-series.

The aim of this paper is to extend the domain of polygamma function from the set of complex numbers to the set of bicomplex numbers. We also discuss integral representation, recurrence relation, multiplication formula and reflection formula for this function.

This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

In this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

The purpose of present paper is to introduce a new extension of Hurwitz-Lerch Zeta function by using the extended Beta function. Some recurrence relations, generating relations and integral representations are derived for that new extension.

Confluent Appell polynomials

New relations and transformation formulas for the Appell function F2 (a, b, b′; c, c′; w, z) and confluent Appell functions (Humbert functions) Ψ1, Ψ2 are obtained. These relations include limit formulas, integral representations, differentiation and recurrence formulas. Summation formulas for F2, Ψ1, and Ψ2 are derived.

In this paper, we present p+q+m and p+q+2m parametric S-functions, exploring the relationships between these two types of functions. We derive their fundamental properties, including generating functions, recurrence relations, differential formulas and integral representations. From these results, we establish several notablecorollaries. Furthermore, we present a distribution function involving the S-function and compute the Laplace transform of its density function. Additionally, we explore the connections between the S-function and other significant transcendental functions.

<abstract><p>In recent years, much attention has been paid to the role of special matrix polynomials of a real or complex variable in mathematical physics, especially in boundary value problems. In this article, we define a new type of matrix-valued polynomials, called the first Appell matrix polynomial of two complex variables. The properties of the newly definite matrix polynomial involving, generating matrix functions, recurrence relations, Rodrigues' type formula and integral representation are investigated. Further, relevant connections between the first Appell matrix polynomial and various matrix functions are reported. The current study may open the door for further investigations concerning the practical applications of matrix polynomials associated with a system of differential equations.</p></abstract>