Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Evaluation of beta integrals of Ramanujan type and integral representations for bilateral hypergeometric series

Howard Cohl, Hans Volkmer

Abstract

In this paper we evaluate integrals of products of gamma functions of Ramanujan type in terms of bilateral hypergeometric series. In cases where the bilateral hypergeometric series are summable, then we evaluate these integral as beta integrals. In addition, we obtain integral representations for bilateral hypergeometric series.

Evaluation of beta integrals of Ramanujan type and integral representations for bilateral hypergeometric series

Abstract

In this paper we evaluate integrals of products of gamma functions of Ramanujan type in terms of bilateral hypergeometric series. In cases where the bilateral hypergeometric series are summable, then we evaluate these integral as beta integrals. In addition, we obtain integral representations for bilateral hypergeometric series.

Paper Structure

This paper contains 19 sections, 26 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on bilateral hypergeometric series
  3. Bilateral hypergeometric series
  4. Basic hypergeometric series
  5. The limit q->1
  6. The Poisson summation formula
  7. Main results
  8. Special cases
  9. The case m=1
  10. The case m=2
  11. The case m=3
  12. The case m=4
  13. The case m=5
  14. The case m=6
  15. A Fourier transform and q-extensions
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\alpha=s+it$ with $s>0$, $t\in \mathbb{R}$. There is a constant $K$ independent of $q$ and $n$ such that

Theorems & Definitions (26)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Lemma 4.1
  • Theorem 4.2
  • Lemma 4.3
  • Theorem 4.4
  • ...and 16 more