Some basic bilateral sums and integrals

Mourad Ismail and Mizan Rahman

Vol. 170 (1995), No. 2, 497–515

DOI: 10.2140/pjm.1995.170.497

Abstract

By splitting the real line into intervals of unit length a doubly infinite integral of the form ∫ _−∞^∞F(q^x)dx, 0 < q < 1, can clearly be expressed as ∫ ₀¹∑ _n=−∞^∞F(q^x+n)dx, provided F satisfies the appropriate conditions. This simple idea is used to prove Ramanujan’s integral analogues of his ₁ψ₁ sum and give a new proof of Askey and Roy’s extention of it. Integral analogues of the well-poised ₂ψ₂ sum as well as the very-well-poised ₆ψ₆ sum are also found in a straightforward manner. An extension to a very-well-poised and balanced ₈ψ₈ series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.

Mathematical Subject Classification 2000

Primary: 33D05

Secondary: 33D20, 33D45

Milestones

Received: 8 February 1993

Revised: 21 January 1994

Published: 1 October 1995