Two-sided inequalities for the extended Hurwitz-Lerch zeta function. (English) Zbl 1228.11137

Srivastava, H. M.; Jankov, Dragana; Pogány, Tibor K.; Saxena, R. K.

Comput. Math. Appl. 62, No. 1, 516-522 (2011).

Summary: Recently, the authors [Integral Transforms Spec. Funct. 22, No. 7, 487–506 (2011; Zbl 1242.11065)] unified and extended several interesting generalizations of the familiar Hurwitz-Lerch zeta function \(\Phi (z,s,a)\) by introducing a Fox-Wright type generalized hypergeometric function in the kernel. For this newly introduced special function, two integral representations of different kinds are investigated here by means of a known result involving a Fox-Wright generalized hypergeometric function kernel, which was given earlier in [loc. cit], and by applying some Mathieu \((\alpha,\lambda)\)-series techniques. Finally, by appealing to each of these two integral representations, two sets of two-sided bounding inequalities are proved, thereby extending and generalizing the recent work by the last three authors [Appl. Math. Lett. 24, No. 8, 1473–1476 (2011; Zbl 1228.11135)].

MSC:

11M35	Hurwitz and Lerch zeta functions
30D15	Special classes of entire functions of one complex variable and growth estimates
33B15	Gamma, beta and polygamma functions
33C20	Generalized hypergeometric series, \({}_pF_q\)

Keywords:

extended Hurwitz-Lerch zeta function; Fox-Wright function; hypergeometric function; Mathieu \((\alpha,\lambda)\)-series techniques; psi function; digamma) function; two-sided bounding inequalities

Citations:

Zbl 1242.11065; Zbl 1228.11135

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