Birth and death processes are useful in a wide range of disciplines from computer networks and telecommunications to chemical kinetics and epidemiology. Data from many different areas such as linguistics, music, or warfare fit Zipf’s law surprisingly well. The Lerch distribution generalizes Zipf’s law and is applicable in survival and dispersal processes. In this article we construct a birth an...

Making use of the Hurwitz–Lerch Zeta function, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients of complex order denoted by TS b (α, β, γ) and obtain coefficient estimates, extreme points, the radii of close to convexity, starlikeness and convexity and neighbourhood results for the class TS b (α, β, γ). In parti...

A family of generalized definite logarithmic integrals given by $$\int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i (a)-i (x))^k\right)}{(x+1)^2}dx$$built over the Lerch function has its analytic properties and special values listed in explicit detail. We use general method as [6] to derive this integral. then give a number examples that can be derived from integral terms well know...

In this paper, we give an explicit description of the complex and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic ...