ARTICLE | doi:10.20944/preprints201803.0008.v1
Subject: Mathematics & Computer Science, Analysis Keywords: Extended beta function, hypergeometric functions, Hurwitz-Lerch Zeta functions, Mellin transform, transformation formula and summation formula.
Online: 1 March 2018 (14:46:40 CET)
In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.
ARTICLE | doi:10.20944/preprints202204.0176.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: ceiling function; floor function; Fibonacci Number; Generalised Dirichlet series; Lerch - Zeta Function; Hurwitz - Zeta function; Polylogarithm; Riemann-Zeta function
Online: 19 April 2022 (06:03:59 CEST)
In this part of the series of two papers, we extend the theorems discussed in part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, Polylogarithm and Fibonacci numbers. In continuation, we obtain some zeros of the newly developed zeta functions and explain their behaviour using plots in complex plane. Furthermore, we provide particular cases for the theorems and corollaries which show that our results generalise the currently available functions and series such as the Riemann zeta function and the geometric series. Finally, we provide four miscellaneous examples to showcase the vast scope of the developed theorems.
ARTICLE | doi:10.20944/preprints202002.0245.v1
Subject: Mathematics & Computer Science, Analysis Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zetafunction; Laurent coefficients
Online: 17 February 2020 (15:10:08 CET)
Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function
ARTICLE | doi:10.20944/preprints201703.0029.v1
Subject: Mathematics & Computer Science, Analysis Keywords: series identity; Catalan number; Catalan function; Riemanian zeta function; alternative Hurwitz zeta function; digamma function
Online: 6 March 2017 (07:01:20 CET)
In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.