Generalized Zeta Functions (original) (raw)
Related papers
Various Series Concerning the Zeta Function
International Journal of Emerging Multidisciplinaries: Mathematics
In this paper we evaluated various series concerning the ζ function. We also have shown how our Lemma can be paired up with different generating functions to produce more series as a consequence.
Certain Classes of Series Involving the Zeta Function
Journal of Mathematical Analysis and Applications, 1999
The authors apply the theory of the double gamma function, which was recently revived in the study of determinants of Laplacians, to evaluate some families of series involving the Riemann zeta function. Introducing a (presumably new) mathematical constant in the theory of the double gamma function, they also systematically evaluate a definite integral of the double gamma function and various classes of series associated with zeta functions. Some of these definite integrals are expressed in terms of quotients of double gamma functions.
Riemann Zeta Function, Hurwitz Zeta Function, Epstein Zeta Function, Mellin Transform
Mathematics
In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials.
A certain family of series associated with the zeta and related functions
Hiroshima Mathematical Journal
The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690-1764) and Leonhard Euler (1707-1783). Many di¤erent techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Zeta function can be evaluated by starting with a single known identity for the generalized (or Hurwitz) Zeta function. Some special cases and their connections with already developed series involving the Zeta and related functions are also considered.
Further generalization of the extended Hurwitz-Lerch Zeta functions
Boletim da Sociedade Paranaense de Matemática, 2017
Recently various extensions of Hurwitz-Lerch Zeta functions have been investigated. Here, we first introduce a further generalization of the extended Hurwitz-Lerch Zeta functions. Then we investigate certain interesting and (potentially) useful properties, systematically, of the generalization of the extended Hurwitz-Lerch Zeta functions, for example, various integral representations, Mellin transform, generating functions and extended fractional derivatives formulas associated with these extended generalized Hurwitz-Lerch Zeta functions. An application to probability distributions is further considered. Some interesting special cases of our main results are also pointed out.
Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function
Journal of Inequalities and Applications, 2013
Many useful and interesting properties, identities, and relations for the Riemann zeta function ζ (s) and the Hurwitz zeta function ζ (s, a) have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen's method. We also present a double inequality approximating ζ (2r + 1) by a more rapidly convergent series.
Four Variants of Riemann Zeta Function
Mathematics
By means of the generating function method and Dougall’s formulae for bilateral hypergeometric series, we examine four classes of infinite series, which may be considered as variants of Riemann zeta function. Several summation formulae are established in closed form, which shows remarkably that the values of these series result in multiples of integer powers of π either by rational numbers or by algebraic numbers.