A unified approach to the summation and integration formulas for q-hypergeometric functions I1

Mizan Rahman, Sergei Suslov

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

The most basic summation formula in the theory of q-hypergeometric functions is the well-known q-binomial formula. Not so well-known is the fact that there is a bilateral extension of it due to Ramanujan, and that there are two integral analogues of it. We show that these summation formulas as well as their integral counterparts have essentially the same origin, namely, a Pearson-type difference equation on a q-linear lattice. It is shown that the boundary conditions determine the structure of the solution of this equation which also enables us to evaluate the sums and integrals by a systematic process of iteration. We conclude by giving a very simple derivation of the q-Gauss formula and a second summation formula for a nonterminating 2φ1 series.

Original languageEnglish (US)
Pages (from-to)101-118
Number of pages18
JournalJournal of Statistical Planning and Inference
Volume54
Issue number1
DOIs
StatePublished - Sep 2 1996
Externally publishedYes

Fingerprint

Summation Formula
Hypergeometric Functions
Difference equations
Summation
Boundary conditions
Ramanujan
Difference equation
Gauss
Analogue
Iteration
Series
Evaluate
Integral
Hypergeometric functions

Keywords

  • Basic hypergeometric series
  • Pearson equation
  • Summation and integration formulas

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Applied Mathematics
  • Statistics and Probability

Cite this

A unified approach to the summation and integration formulas for q-hypergeometric functions I1 . / Rahman, Mizan; Suslov, Sergei.

In: Journal of Statistical Planning and Inference, Vol. 54, No. 1, 02.09.1996, p. 101-118.

Research output: Contribution to journalArticle

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