A new three-dimensional integral containing f(x,y,z)Iv(x?) is derived where Iv(x?) the Modified Bessel Function of first kind and taken over infinite cubic space 0<x<?,0<y<?,0<z<?. The not easily evaluated for complex ranges parameters. representation in terms Hurwitz–Lerch zeta function, polylogarithm function Riemann functions are evaluated. This yields triple representation...

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by A.P. Guinand and as a limiting case of the second formu...

The behaviour of the real roots of the Bernoulli polynomials Bm(a) for large m is investigated. It is shown that if N(m) is the number of these roots then lim m→∞ N(m) m = 2 πe We also show that on the interval Im : − [ m−2 2πe ] < a < 1 + [ m−2 2πe ] , the roots of Bm(a) are close to the half-integer lattices: a = 0,±1/2,±1,±3/2,±2, ... if m is odd, and a = ±1/4,±3/4,±5/4,±7/4, ... if m is eve...

In this paper, which is a companion paper to W , starting from the Euler integral which appears in a generalization of Jensen’s formula, we shall give a closed form for the integral of log Γ 1± t . This enables us to locate the genesis of two new functions A1/a and C1/a considered by Srivastava and Choi. We consider the closely related function A(a) and the Hurwitz zeta function, which render t...

In this paper we construct trigonometric functions of the sum Tp(h, k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric representations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations between the Clausen functions, Polylogarithm function, Hurwitz zeta function, generalized La...

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...