Hurwitz-Lerch Zeta Function Research Articles

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.

New results for Srivastava’s λ-generalized Hurwitz-Lerch Zeta function

In view of the relationship with the Kr?tzel function, we derive a new series representation for the ?-generalized Hurwitz-Lerch Zeta function introduced by H.M. Srivastava [Appl. Math. Inf. Sci. 8 (2014) 1485-1500] and determine the monotonicity of its coeficients. An integral representation of the Mathieu (a;?)-series is rederived by applying the Abel?s summation formula (which provides a slight modification of the result given by Pog?ny [Integral Transforms Spec. Funct. 16 (8) (2005) 685-689]) and this modified form of the result is then used to obtain a new integral representation for Srivastava?s ?-generalized Hurwitz-Lerch Zeta function. Finally, by making use of the various results presented in this paper, we establish two sets of two-sided inequalities for Srivastava?s ?-generalized Hurwitz-Lerch Zeta function.

Univalency and starlikeness of normalized Hurwitz-Lerch Zeta function inside unit disk

Univalency and starlikeness of normalized Hurwitz-Lerch Zeta function inside unit disk

On extended Hurwitz–Lerch zeta function

This paper investigates an extended form of a beta function Bp,q(x,y). We first study the convergence problem of the function Bp,q(x,y) and consider the completely monotonic and log-convex properties of this function. As a result, we obtain a pair of Laguerre type inequalities. Next, we provide a new double integral representation for the function Bp,q(x,y). Subsequently, we consider the convergence problem of the extended Hurwitz–Lerch zeta function Φλ,μ;ν(z,s,a;p,q) defined by its series representation. Upon using the series manipulation techniques, we obtain two series identities. We also find various integral representations for the function Φλ,μ;ν(z,s,a;p,q). Lastly, we apply Fourier analysis to the function zaΦμ;ν(z,s,a;p,q) and obtain a Lindelöf–Wirtinger type expansion. Some interesting and promising results are also illustrated.

Certain problems related to generalized Srivastava–Attiya operator

In this paper, we study certain properties involving the generalized Srivastava–Attiya operator. A set of subordination results are obtained and some special cases connected with the Hurwitz–Lerch zeta function and their relevances with known results are also pointed out.

Real zeros of Hurwitz–Lerch zeta functions in the interval (−1,0)

For 0<a≤1, s,z∈C and 0<|z|≤1, the Hurwitz–Lerch zeta function is defined by Φ(s,a,z):=∑n=0∞zn(n+a)−s when σ:=ℜ(s)>1. In this paper, we show that Φ(σ,a,z)≠0 when σ∈(−1,0) if and only if [I] z=1 and (3−3)/6≤a≤1/2 or (3+3)/6≤a≤1, [II] z∈[−1,1) and (1−z)(1−a)≤1, [III] z∉R and 0<a≤1. In addition, we give a new proof of the functional equation of Φ(s,a,z).

BI-UNIVALENT FUNCTIONS OF COMPLEX ORDER BASED ON SUBORDINATE CONDITIONS INVOLVING HURWITZ-LERCH ZETA FUNCTION

The purpose of the present paper is to introduce and investigate two new subclasses of bi-univalent functions of complex order defined in the open unit disk, which are associated with Hurwitz-Lerch zeta function and satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients <TEX>${\mid}a_2{\mid}$</TEX> and <TEX>${\mid}a_3{\mid}$</TEX> for functions in the new subclasses. Several (known or new) consequences of the results are also pointed out.

New Results Involving a Class of Generalized Hurwitz-Lerch Zeta Functions and Their Applications

In this paper, we study a certain class of generalized Hurwitz-Lerch zeta functions. We derive several new and useful properties of these generalized Hurwitz-Lerch zeta functions such as (for example) their partial differential equations, new series and Mellin-Barnes type contour integral representations involving Fox’s H-function and a pair of summation formulas. More importantly, by considering their application in Number Theory, we construct a new continuous analogue of Lippert’s Hurwitz measure. Some statistical applications are also given.

Some subclasses of meromorphic functions involving in the Hurwitz-Lerch Zeta function

The main purpose of this paper is to investigate some subclasses of meromorphic functions involving the meromorphic modified version of the familiar Srivastava-Attiya operator. Such results as inclusion relationships, convolution properties, coefficient inequalities, integralpreserving properties, subordination and superordination properties are proved.

An integrodifferential operator for meromorphic functions associated with the Hurwitz-Lerch zeta function

In this paper, we introduce a new integrodifferential operator associated with the Hurwitz Lerch Zeta function in the puncture open disk of the meromorphic functions. We also obtain some properties of the third-order differential subordination and superordination for this integrodifferential operator, by using certain classes of admissible functions.

Generalizations of poly-Bernoulli and poly-Cauchy numbers

In this paper we consider some generalizations of poly-Bernoulli and poly-Cauchy numbers. The first is by means of the Hurwitz–Lerch zeta function. The second generalization is via weighted Stirling numbers. The third one is given with the help of degenerate Stirling numbers. All these generalizations lead to symmetries between various types of Stirling numbers, and enable us to investigate and expand algebraic properties of poly-Bernoulli and poly-Cauchy numbers. We also combine these generalizations and derive numerous combinatorial and arithmetical identities.

Strong Differential Subordination and Superordination for Multivalently Meromorphic Functions Involving the Hurwitz–Lerch Zeta Function

An integral operator for meromorphic multivalent function in the punctured unit disk is introduced by means of Hurwitz–Lerch Zeta function. Strong differential subordination and superordination results associated with the operator are obtained by investigating appropriate classes of admissible functions.

Partial Sums of Certain Classes of Meromorphic Functions Related to the Hurwitz-Lerch Zeta Function

AbstractIn the present paper, we give sufficient conditions for a function f to be in the subclasses ΣS*a,s(A. B, α, β) and ΣKa,s(A, B, α, β) of the class Σ of meromorphic functions which are analytic in the punctured unit disk U*. We further investigate the ratio of a function related to the Hurwitz-Lerch zeta function and its sequence of partial sums.

On the functional equation of the normalized Shintani L-function of several variables

In this paper, we introduce the normalized Shintani L-function of several variables by an integral representation and prove its functional equation. The Shintani L-function is a generalization to several variables of the Hurwitz–Lerch zeta function and the functional equation given in this paper is a generalization of the functional equation of Hurwitz–Lerch zeta function. In addition to the functional equation, we give special values of the normalized Shintani L-function at non-positive integers and some positive integers.

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.

Some new results related to a class of generalized Hurwitz zeta function

In this paper, we establish some new results associated with a class of functions (re- lated to Hurwitz-Lerch zeta function) defined and introduced by Raina and Chhajed in (Acta Math. Univ. Comenianae, 73 (2004), 89-100). Among the results obtained are the series repre- sentation, generating function relationship and their corresponding multidimensional extensions. Some reduced cases of our results are also discussed.

Applications of the Hurwitz-Lerch Zeta-Function

In this paper, we shall exhibit the use of two principles, “principle of decomposition into residue classes” and “binomial principle of analytic continuation” due to Ram Murty and Sinha and indicate a certain distribution property and the functional equation for the Lipschitz-Lerch transcendent at integral arguments ofs. By considering the limiting cases ,we can also deduce new striking identities for Lipschizt-Lerch transcendent, among which is the Gauss second formula for the digamma function, Lipschitz-Lerch transcendent

Certain subclasses of meromorphically univalent functions defined by a linear operator associated with the λ-generalized Hurwitz–Lerch zeta function

By using a linear operator associated with the λ-generalized Hurwitz–Lerch zeta function, which is defined here by means of the Hadamard product (or convolution), the authors introduce and investigate various properties of certain subclasses of meromorphically univalent functions in the punctured unit disk. Some closely related (known or new) corollaries and consequences of the main results presented in this paper are also considered.

Further results involving a class of generalized Hurwitz-Lerch zeta functions

In this paper, we present several new expansion formulas for a class of generalized Hurwitz-Lerch zeta functions which were introduced by Raina and Chhajed [R. K. Raina and P. K. Chhajed, “Certain Results Involving a Class of Functions Associated with the Hurwitz Zeta Function,” Acta Math. Univ. Comenian. 73, 89–100 (2004)] and (more recently) by Srivastava et al. [H. M. Srivastava, M.-J. Luo, and R. K. Raina, “New Results Involving a Class of Generalized Hurwitz-Lerch Zeta Functions and Their Applications,” Turkish J. Anal. Number Theory 1, 26–35 (2013)]. These expansion formulas are obtained with the help of some fractional calculus theorems such as the generalized Leibniz rules, the Taylorlike expansions in terms of different functions and the generalized chain rule. Several (known or new) special cases are also considered.

A New Family of the λ -Generalized Hurwitz-Lerch Zeta Functions with Applications

Motivated largely by a number of recent investigations, we introduce and investigate the various properties of a certain new family of the λ -generalized Hurwitz-Lerch zeta functions. We derive many potentially useful results involving these λ -generalized Hurwitz-Lerch zeta functions including (for example) their partial differ ential equations, new series and Mellin-Barnes type contour integral representations (which are associated with Fox's H-function) and several other summation formulas. We discuss their potential application in Number Theory by appropriately constructing a seemingly novel continuous analogue of Lippert's Hurwitz measure. We also consider some other statistical applications of the family of the λ -generalized Hurwitz-Lerch zeta functions in probability distribution theory.