Generalized Hurwitz-Lerch Zeta Function Research Articles

The study of generalized Hurwitz–Lerch zeta function and fractional kinetic equations

The study of generalized Hurwitz–Lerch zeta function and fractional kinetic equations

New results for generalized Hurwitz-Lerch Zeta functions using Laplace transform

Abstract Fractional Kinetic equations (FKEs) including a wide variety of special functions are widely and successfully applied in describing and solving many important problems of physics and astrophysics. In this work, the solutions of the FKEs of the generalized Hurwitz-Lerch Zeta function using the Laplace transform are derived and examined.

An Operator Defined on Hadamard Product Pertaining to Generalized Hurwitz-Lerch Zeta Function with Conical Section

The author’s goal is to highlight the most recent developments in the research on the study of complex-valued functions as seen through an understanding of geometric function theory. Contributions will be required for any aspect of the Hadamard product associated to the generalized Hurwitz-Lerch Zeta function convoluted with the theory of functions that are univalent. In the present investigation, the author focused into the inclusion relations of a few subclasses of the -starlike functions, -uniformly convex functions, and -quasi-convex functions which together make up the generalized Hurwitz-Lerch Zeta function.

Solution of fractional kinetic equations involving generalized Hurwitz-Lerch Zeta function using Sumudu Transform

Fractional kinetic equations (FKEs) comprising a large array of special functions have been extensively and successfully applied in specification and solving many significant problems of astrophysics and physics. In this present work, our aim is to demonstrate solutions of (FKEs) of the generalized Hurwitz-Lerch Zeta function by applying the Sumudu transform. In addition to these, solutions of (FKEs) in special conditions of generalised Hurwitz-Lerch Zeta function have been derived.

Some families of the general Mathieu‐type series with associated properties and functional inequalities

The main purpose of this paper is to introduce general families of the extended Mathieu‐type power series and present a number of potentially useful integral representations of several general families of the extended Mathieu‐type power series in a unified manner. Relationships of the extended Mathieu‐type functional power series with the generalized Hurwitz–Lerch zeta function is also considered. Various other properties, mainly, Mellin transform and Hankel transform, and fractional derivative formulae are derived for the extended Mathieu series. A pair of the bounding inequalities are established for the extended Mathieu‐type series. As an application of newly defined function, we present a systematic study of probability density function and distribution function associated with the general extended Mathieu‐type power series. In particular, the mathematical expectation and variance of the distribution are derived. Finally, we prove some properties of monotonicity, convexity, and Turán‐type inequalities for the general extended Mathieu‐type power series.

Note on the Hurwitz–Lerch Zeta Function of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

On The Third-Order Complex Differential Inequalities of ξ-Generalized-Hurwitz–Lerch Zeta Functions

In the z- domain, differential subordination is a complex technique of geometric function theory based on the idea of differential inequality. It has formulas in terms of the first, second and third derivatives. In this study, we introduce some applications of the third-order differential subordination for a newly defined linear operator that includes ξ -Generalized-Hurwitz–Lerch Zeta functions (GHLZF). These outcomes are derived by investigating the appropriate classes of admissible functions.

Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

Remarks on a main inequality for several special functions

ABSTRACTIt is shown that the main inequality for several special functions derived in [Masjed-Jamei M. A main inequality for several special functions. Comput Math Appl. 2010;60:1280–1289] can be put in a concise form, and that the main inequalities of the first kind Bessel function, Laplace and Fourier transforms are not valid as presented in the aforementioned paper. To provide alternative inequalities, we give a generalization, being in some cases an improvement, of the Cauchy–Bunyakovsky–Schwarz inequality which can be applied to real functions not necessarily of constant sign. The corresponding discrete inequality is also obtained, which we use to improve the inequalities of the Riemann zeta and the generalized Hurwitz–Lerch zeta functions. Finally, from the main concise inequality, we derive a Turán-type inequality.

An extension of the generalized Hurwitz-Lerch Zeta function of two variables

The main object of this paper is to introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate such its several interesting properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results, a summation formula and Mellin-Barnes type contour integral representations. We also consider some important special cases.

Some further properties of a linear operator associated with the [formula omitted]-generalized Hurwitz–Lerch zeta function related to the class of meromorphically univalent functions

In this sequel to the recent work (see Srivastava, 2015), we investigate some further properties of a linear operator associated with Srivastava’s λ-generalized Hurwitz–Lerch zeta function, which are related to a certain class of meromorphically univalent functions in the punctured unit disk defined here by means of the Hadamard product (or convolution). We also give several interesting corollaries and consequences of the main results presented in this paper.

Srivastava-Pintér theorems for 2D-Appell polynomials and their applications

Recently, Srivastava and Pinter proved addition theorems for the generalized Bernoulli and Euler polynomials. Luo and Srivastava obtained the anologous results for the generalized Apostol–Bernoulli polynomials and the generalized Apostol–Euler polynomials. Finally, Tremblay et al. gave analogues of the Srivastava–Pinter addition theorem for general family of Bernoulli polynomials. In this paper, we obtain Srivastava–Pinter type theorems for 2D-Appell Polynomials. We also give the representation of 2D-Appell Polynomials in terms of the Stirling numbers of the second kind and 1D-Appell polynomials. Furthermore, we introduce the unified 2D-Apostol polynomials. In particular, we obtain some relations between that family of polynomials and the generalized Hurwitz–Lerch zeta function as well as the Gauss hypergeometric function. Finally, we present some applications of Srivastava–Pinter type theorems for 2D-Appell Polynomials. Copyright © 2013 John Wiley & Sons, Ltd.

SPECIAL VALUES OF SOME EXTENSIONS OF ZETA FUNCTION VIA BETA DISTRIBUTIONS

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

Integral and computational representations of the extended Hurwitz–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.

Notes on generalization of the Bernoulli type polynomials

Abstract Recently, Srivastava et al. introduced a new generalization of the Bernoulli, Euler and Genocchi polynomials (see [H.M. Srivastava, M. Garg, S. Choudhary, Russian J. Math. Phys. 17 (2010) 251–261] and [H.M. Srivastava, M. Garg, S. Choudhary, Taiwanese J. Math. 15 (2011) 283–305]). They established several interesting properties of these general polynomials, the generalized Hurwitz–Lerch zeta functions and also in series involving the familiar Gaussian hypergeometric function. By the same motivation of Srivastava’s et al. [11,12] , we introduce and derive multiplication formula and some identities related to the generalized Bernoulli type polynomials of higher order associated with positive real parameters a , b and c . We also establish multiple alternating sums in terms of these polynomials. Moreover, by differentiating the generating function of these polynomials, we give a interpolation function of these polynomials.

Some New Families of Generalized Euler and Genocchi Polynomials

The main object of this paper is to introduce and investigate a new generalization of the family of Euler polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials and derive explicit representations for them in terms of a certain generalized Hurwitz-Lerch Zeta function and in series involving the familiar Gaussian hypergeometric function. Finally, we propose an analogous generalization of the closely-related Genocchi polynomials and show how we can fruifully exploit some potentially useful linear connections of all these three important families of generalized Bernoulli, Euler and Genocchi polynomials with one another.

Some formulas for Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments

We give some explicit relationships between the Apostol-Euler polynomials and generalized Hurwitz-Lerch Zeta function and obtain some series representations of the Apostol-Euler polynomials of higher order in terms of the generalized Hurwitz-Lerch Zeta function. Several interesting special cases are also shown.

Some expansion formulas for a class of generalized Hurwitz–Lerch Zeta functions

By making use of fractional calculus, the authors present a systematic investigation of expansion and transformation formulas for several general families of the Hurwitz–Lerch Zeta functions. Relevant connections of the results discussed here with those obtained in earlier works are also indicated precisely.