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On a generalization of the Rogers generating function.

Howard S Cohl1, Roberto S Costas-Santos2, Tanay V Wakhare3

  • 1Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA.

Journal of Mathematical Analysis and Applications
|December 29, 2022
PubMed
Summary
This summary is machine-generated.

This study generalizes Rogers generating functions for continuous q-ultraspherical polynomials, yielding new expansions for several polynomial families. A key finding is a novel quadratic transformation for basic hypergeometric functions.

Keywords:
Basic hypergeometric orthogonalBasic hypergeometric seriesConnection coefficientsDefinite integralsEigenfunction expansionsGenerating functionspolynomials

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Area of Science:

  • Mathematical Physics
  • Special Functions
  • Orthogonal Polynomials

Background:

  • The study builds upon existing work on Rogers generating functions and their generalizations.
  • It addresses the need for unified expansions of various orthogonal polynomials.

Purpose of the Study:

  • To derive generalized Rogers generating functions for continuous q-ultraspherical/Rogers polynomials.
  • To obtain new expansions for continuous q-Hermite, q-Legendre, Laguerre, and Chebyshev polynomials.
  • To establish a new quadratic transformation for basic hypergeometric functions.

Main Methods:

  • Derivation of generalized Rogers generating functions.
  • Utilizing coefficient comparison between different polynomial expansions.
  • Application of orthogonality for integral representations.

Main Results:

  • A generalized Rogers generating function with a 2Φ1 coefficient.
  • Expansions for continuous q-Hermite, q-Legendre, Laguerre, and Chebyshev polynomials.
  • A new quadratic transformation relating 8Φ7 to 2Φ1 basic hypergeometric functions.
  • Definite integral representations for the derived expansions.

Conclusions:

  • The derived expansions provide a unified framework for several important polynomial families.
  • The new quadratic transformation offers a significant advancement in the theory of basic hypergeometric functions.
  • The integral representations offer new tools for analyzing these polynomials.