### 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 _{2}φ_{1} 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 | |

State | Published - Sep 2 1996 |

Externally published | Yes |

### Fingerprint

### 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 I ^{1}
.** / Rahman, Mizan; Suslov, Sergei.

Research output: Contribution to journal › Article

^{1}',

*Journal of Statistical Planning and Inference*, vol. 54, no. 1, pp. 101-118. https://doi.org/10.1016/0378-3758(95)00160-3

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TY - JOUR

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

AU - Rahman, Mizan

AU - Suslov, Sergei

PY - 1996/9/2

Y1 - 1996/9/2

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

UR - http://www.scopus.com/inward/record.url?scp=0030565345&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030565345&partnerID=8YFLogxK

U2 - 10.1016/0378-3758(95)00160-3

DO - 10.1016/0378-3758(95)00160-3

M3 - Article

AN - SCOPUS:0030565345

VL - 54

SP - 101

EP - 118

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 1

ER -