Research Article
New Relations Involving an Extended Multiparameter
Hurwitz-Lerch Zeta Function with Applications
H. M. Srivastava,
1
Sébastien Gaboury,
2
and Richard Tremblay
2
1
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4
2
Department of Mathematics and Computer Science, University of Quebec at Chicoutimi, Chicoutimi, QC, Canada G7H 2B1
Correspondence should be addressed to S´ ebastien Gaboury; s1gabour@uqac.ca
Received 28 February 2014; Accepted 16 April 2014; Published 13 May 2014
Academic Editor: Shamsul Qamar
Copyright © 2014 H. M. Srivastava et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We derive several new expansion formulas involving an extended multiparameter Hurwitz-Lerch zeta function introduced and
studied recently by Srivastava et al. (2011). Tese expansions are obtained by using some fractional calculus methods such as the
generalized Leibniz rules, the Taylor-like expansions in terms of diferent functions, and the generalized chain rule. Several (known
or new) special cases are also given.
1. Introduction
Te Hurwitz-Lerch zeta function Φ(,,) which is one of the
fundamentally important higher transcendental functions is
defned by (see, e.g., [1, page 121 et seq.]; see also [2] and [3,
page 194 et seq.])
Φ(,,):=
∞
∑
=0
( + )
,
( ∈ C \ Z
−
0
;∈ C when || < 1;
R () > 1 when || = 1) .
(1)
Te Hurwitz-Lerch zeta function contains, as its special cases,
the Riemann zeta function (), the Hurwitz zeta function
(, ), and the Lerch zeta function ℓ
() defned by
() :=
∞
∑
=1
1
=Φ(1,,1)=(,1) (R () > 1) , (2)
(, ) :=
∞
∑
=0
1
( + )
=Φ(1,,)
(R () > 1; ∈ C \ Z
−
0
),
(3)
ℓ
() :=
∞
∑
=1
2
( + 1)
=Φ(
2
,,1) (R () > 1; ∈ R),
(4)
respectively.
Te Hurwitz-Lerch zeta function is connected with other
special functions of analytic number theory such as the
polylogarithmic function (or de Jonqui` ere’s function)
():
() :=
∞
∑
=1
=Φ(,,1)
( ∈ C when || < 1; R () > 1 when || = 1)
(5)
and the Lipschitz-Lerch zeta function (,,) (see [1, page
122, Equation 2.5 (11)])
(,,):=
∞
∑
=0
2
( + )
=Φ(
2
,,)
( ∈ C \ Z
−
0
; R () > 0 when ∈ R \ Z;
R () > 1 when ∈ Z).
(6)
Te Hurwitz-Lerch zeta function Φ(,,) defned in (7) can
be continued meromorphically to the whole complex -plane,
Hindawi Publishing Corporation
International Journal of Analysis
Volume 2014, Article ID 680850, 14 pages
http://dx.doi.org/10.1155/2014/680850