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.

[1]  M. A. Chaudhry,et al.  On a Class of Incomplete Gamma Functions with Applications , 2001 .

[2]  Shy-Der Lin,et al.  Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations , 2004, Appl. Math. Comput..

[3]  Nobushige Kurokawa,et al.  Number Theory 2: Introduction to Class Field Theory , 2011 .

[4]  A. W. Kemp,et al.  A treatise on generating functions , 1984 .

[5]  H. M.  Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions , 2013, SpringerPlus.

[6]  Hari M. Srivastava,et al.  Series Associated with the Zeta and Related Functions , 2001 .

[7]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[8]  George B. Arfken,et al.  More Special Functions , 2012 .

[9]  S. Golomb.  A class of probability distributions on the integers , 1970 .

[10]  Arak M. Mathai,et al.  The H-Function with Applications in Statistics and Other Disciplines. , 1981 .

[11]  A. M. Mathai,et al.  Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences , 1973 .

[12]  H. Srivastava,et al.  Zeta and q-Zeta Functions and Associated Series and Integrals , 2011 .

[13]  A. Kilbas.  H-Transforms: Theory and Applications , 2004 .

[14]  A. M. Mathai,et al.  The H-Function: Theory and Applications , 2009 .

[15]  A. Erdélyi,et al.  Tables of integral transforms , 1955 .

[16]  R. K. Raina,et al.  CERTAIN RESULTS INVOLVING A CLASS OF FUNCTIONS ASSOCIATED WITH THE HURWITZ ZETA FUNCTION , 2004 .

[17]  Hari M. Srivastava,et al.  Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler , 2011 .

[18]  Hari M. Srivastava,et al.  The H-functions of one and two variables, with applications , 1982 .

[19]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[20]  Hari M. Srivastava,et al.  Some formulas for the Bernoulli and Euler polynomials at rational arguments , 2000, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .