TL;DR
This paper provides elementary proofs for the Askey--Wilson integral and its generalizations, introduces two new similar integrals, and extends a transformation formula involving basic hypergeometric series.
Contribution
It offers new elementary proofs, introduces two novel Askey--Wilson type integrals, and generalizes a transformation formula with $_{3}\phi_{2}$ series.
Findings
Elementary proof of Askey--Wilson and Liu's integrals
Introduction of two new Askey--Wilson type integrals
Generalization of a transformation formula with $_{3}\phi_{2}$ series
Abstract
The Askey--Wilson integral is very important in the theory of orthogonal polynomials. Liu's integral is a generalization of the Askey--Wilson integral with many parameters. With the help of the series rearrangement method, we give the elementary proof of them. Furthermore, we establish two new Askey--Wilson type integrals in the similar way and find a generalization of a known transformation formula containing three series.
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Taxonomy
TopicsMathematical functions and polynomials · Advanced Mathematical Identities · Fractional Differential Equations Solutions