Proceedings of the Japan Academy, Series A, Mathematical Sciences

Ramanujan’s $_{1}\psi_{1}$ summation theorem —perspective, announcement of bilateral $q$-Dixon–Anderson and $q$-Selberg integral extensions, and context—

Masahiko Ito and Peter J. Forrester

Full-text: Open access


The Ramanujan $_{1} \psi_{1}$ summation theorem is studied from the perspective of Jackson integrals, $q$-difference equations and connection formulae. This is an approach which has previously been shown to yield Bailey’s very-well-poised $_{6} \psi_{6}$ summation. Bilateral Jackson integral generalizations of the Dixon–Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_{1} \psi_{1}$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulae for Jackson integrals relating to other roots systems obtained from the same perspective.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 7 (2014), 92-97.

First available in Project Euclid: 7 August 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33D15: Basic hypergeometric functions in one variable, $_r\phi_s$ 33D67: Basic hypergeometric functions associated with root systems
Secondary: 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx]

Dixon–Anderson integral Selberg integral Ramanujan’s $_{1}\psi_{1}$ summation formula Bailey’s very-well-poised $_{6}\psi_{6}$ summation formula


Ito, Masahiko; Forrester, Peter J. Ramanujan’s $_{1}\psi_{1}$ summation theorem —perspective, announcement of bilateral $q$-Dixon–Anderson and $q$-Selberg integral extensions, and context—. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 7, 92--97. doi:10.3792/pjaa.90.92.

Export citation