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

Mizan Rahman; Sergeǐ K. Suslov

doi:10.1016/0378-3758(95)00160-3

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

Mizan Rahman, Sergeǐ K. Suslov

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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 language	English (US)
Pages (from-to)	101-118
Number of pages	18
Journal	Journal of Statistical Planning and Inference
Volume	54
Issue number	1
DOIs	https://doi.org/10.1016/0378-3758(95)00160-3
State	Published - Sep 2 1996
Externally published	Yes

Keywords

Basic hypergeometric series
Pearson equation
Summation and integration formulas

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty
Applied Mathematics

Access to Document

10.1016/0378-3758(95)00160-3

Cite this

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title = "A unified approach to the summation and integration formulas for q-hypergeometric functions I1",

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.",

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author = "Mizan Rahman and Suslov, {Sergeǐ K.}",

note = "Funding Information: * Corresponding author. J This work, supported in part by the NSERC grant # A6197, was completed while the second author was visiting Carleton University in September~October 1993.",

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T1 - A unified approach to the summation and integration formulas for q-hypergeometric functions I1

AU - Rahman, Mizan

AU - Suslov, Sergeǐ K.

N1 - Funding Information: * Corresponding author. J This work, supported in part by the NSERC grant # A6197, was completed while the second author was visiting Carleton University in September~October 1993.

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N2 - 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.

AB - 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.

KW - Basic hypergeometric series

KW - Pearson equation

KW - Summation and integration formulas

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