Some operator identities and q-series transformation formulas

Zhi Guo Liu

doi:10.1016/S0012-365X(02)00626-X

Some operator identities and q-series transformation formulas

Zhi Guo Liu^*

^*Corresponding author for this work

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

105 Scopus citations

Abstract

In this paper, we show how to use the q-exponential operator techniques to derive a transformation formula for the q-Hahn polynomials from the q-Chu-Vandermonde identity. With the same method we also show that the two terms ₃φ ₂ transformation formula of Sears can be recovered from Rogers' iteration of Heine's transformation formula, and the celebrated Sears ₄φ ₃ transformation formula can be derived from his ₃φ ₂ transformation formula with the same method. We also provide new proofs of the three terms Sears ₃φ ₂ transformation formula and an identity of Andrews by our method. We re-derive the q-analogue of Barnes' second lemma from the q-analogue of Barnes' first lemma in one step. In addition we generalize two Ramanujan's formulas for beta integrals as two more general integrals. Finally, we establish two general transformation formulas for bilateral series.

Original language	English
Pages (from-to)	119-139
Number of pages	21
Journal	Discrete Mathematics
Volume	265
Issue number	1-3
DOIs	https://doi.org/10.1016/S0012-365X(02)00626-X
State	Published - 6 Apr 2003
Externally published	Yes

Keywords

Andrews' identity
Barnes' lemma
Bilateral series
Operator identity
Ramanujan's beta integral
Sears' transformation
Transformation formula of q-series
q-Series

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10.1016/S0012-365X(02)00626-X

Cite this

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title = "Some operator identities and q-series transformation formulas",

abstract = "In this paper, we show how to use the q-exponential operator techniques to derive a transformation formula for the q-Hahn polynomials from the q-Chu-Vandermonde identity. With the same method we also show that the two terms 3φ 2 transformation formula of Sears can be recovered from Rogers' iteration of Heine's transformation formula, and the celebrated Sears 4φ 3 transformation formula can be derived from his 3φ 2 transformation formula with the same method. We also provide new proofs of the three terms Sears 3φ 2 transformation formula and an identity of Andrews by our method. We re-derive the q-analogue of Barnes' second lemma from the q-analogue of Barnes' first lemma in one step. In addition we generalize two Ramanujan's formulas for beta integrals as two more general integrals. Finally, we establish two general transformation formulas for bilateral series.",

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AB - In this paper, we show how to use the q-exponential operator techniques to derive a transformation formula for the q-Hahn polynomials from the q-Chu-Vandermonde identity. With the same method we also show that the two terms 3φ 2 transformation formula of Sears can be recovered from Rogers' iteration of Heine's transformation formula, and the celebrated Sears 4φ 3 transformation formula can be derived from his 3φ 2 transformation formula with the same method. We also provide new proofs of the three terms Sears 3φ 2 transformation formula and an identity of Andrews by our method. We re-derive the q-analogue of Barnes' second lemma from the q-analogue of Barnes' first lemma in one step. In addition we generalize two Ramanujan's formulas for beta integrals as two more general integrals. Finally, we establish two general transformation formulas for bilateral series.

KW - Andrews' identity

KW - Barnes' lemma

KW - Bilateral series

KW - Operator identity

KW - Ramanujan's beta integral

KW - Sears' transformation

KW - Transformation formula of q-series

KW - q-Series

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DO - 10.1016/S0012-365X(02)00626-X

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VL - 265

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JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1-3

ER -

Some operator identities and q-series transformation formulas

Abstract

Keywords

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