Abstract
The purpose of this paper is to introduce and investigate a new class of multiple zeta functions of variables. We study its properties, integral representations, differential relation, series expansion and discuss the link with known results.
Highlights
Φ(1, s,1) = ζ (s), (1.6)Φ(1, s,a) = ζ (s,a). (1.7)The generalized (Hurwitz’s) zeta function is defined byThe zeta functions in (1.3) and (1.4) have since been [1,2]extended and generalized by a number of workers ∞ζ (s, a= ) ∑ (a + n)−s (a ≠ 0, −1, −2,...;R(s) > 1), (1.1)n=0 so that when a = 1, we have
In the present paper we introduce a new class of zeta
Functions ζnμ,s1,...,sn ( x1,..., xn ; a1,..., an ) which is defined by ζnμ,s1,...,sn ( x1,..., xn ; a1,..., an )
Summary
Functions ζnμ,s1,...,sn ( x1,..., xn ; a1,..., an ) which is defined by ζnμ,s1,...,sn ( x1,..., xn ; a1,..., an ). ∑ A generalization of (= 1.3) is the Zeta function Φ*μ which is defined by [[3], p.100, (1.5)]:. In view of (2.2), it is seen that where FA(n) is the Lauricella function of nn variables defined by the series By using the Hankel's contour integral for Gamma led to the left-hand side of the formula (2.3). By using the contour integral formula [[2], p.14 (4)]:. We can derive the following interesting formula. One can derive the following contour integral representation. The result follow directly from definition (1.9) and the integral representation (1.11)
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