PPubMed25 Feb 2013Cited 34×

Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions

Hari M Srivastava

Abstract

Title: 2010 Mathematics Subject Classification Primary 11M25, 33C60; Secondary 33C05

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The paper defines extended multi‑parameter Hurwitz‑Lerch Zeta functions and derives new generating‑function and integral‑representation formulas.

What’s new

i.Introduced a family of extended (multi‑parameter) Hurwitz‑Lerch Zeta functions.
ii.Derived generating functions and partial‑sum relations for these extended functions.
iii.Provided several new integral representation formulas (Theorems 1‑6) including sum‑integral, Hankel‑type contour, and Eulerian Beta‑function integrals.
iv.Showed that the τ‑generalized Riemann Zeta function is a notational variant of the standard Hurwitz‑Lerch Zeta function.

How they did it

The author extended existing definitions of the Hurwitz‑Lerch Zeta function, then used series expansions, Mellin‑Barnes contour integrals, and Eulerian Beta‑function integrals to derive generating‑function identities and various integral representations.

What they found

Theorem 1: Integral representations (3.10) and (3.11) for Ω_λ and Θ functions.
Theorem 2: Hankel‑type contour integral formulas (3.17) and (3.18).
Theorem 3: Eulerian Beta‑function integral formulas (3.21) and (3.22).
Theorem 4: Single and double Eulerian Gamma‑function integrals (3.23)–(3.25).
Theorem 6: General sum‑integral representation (4.13) for the extended family.

Caveats

✕All integral formulas require convergence conditions such as min{Re(a),Re(s)} > 0.
✕The τ‑generalizations do not extend beyond the case τ = 1; they are merely notational variations.
✕Some previously published integral formulas (e.g., in Bin‑Saad) are shown to be divergent.

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