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Some Generating Functions for q-Polynomials.

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.

Symmetry
|December 29, 2022
PubMed
Summary
This summary is machine-generated.

This study reveals q-analogues for Bateman, Pasternack, Sylvester, and Cesàro polynomials, establishing a symmetry between calculus and q-calculus. New q-generating functions and classical generating functions for Pasternack and Bateman polynomials are derived.

Keywords:
33C2033D45basic hypergeometric functionsgenerating functionsq-polynomials

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Area of Science:

  • Mathematical Analysis
  • Special Functions
  • q-Calculus

Background:

  • Calculus and q-calculus exhibit underlying symmetries.
  • Classical polynomials like Bateman, Pasternack, Sylvester, and Cesàro have established importance.
  • Generating functions are crucial tools for studying polynomial families.

Purpose of the Study:

  • To establish q-analogues for Bateman, Pasternack, Sylvester, and Cesàro polynomials.
  • To derive q-analogues of generating functions for these polynomials.
  • To discover new classical generating functions for Pasternack and Bateman polynomials.

Main Methods:

  • Utilizing the symmetry between calculus and q-calculus.
  • Deriving q-analogues using basic hypergeometric series (e.g., 4ϕ5, 5ϕ5) and q-Pochhammer symbols.
  • Employing the newly derived q-generating functions to find classical generating functions.

Main Results:

  • Obtained q-analogues of Bateman, Pasternack, Sylvester, and Cesàro polynomials.
  • Derived q-analogues of generating functions expressed via basic hypergeometric series and q-Pochhammer symbols.
  • Identified novel classical generating functions for Pasternack and Bateman polynomials.

Conclusions:

  • The study successfully demonstrates the q-analogue framework for classical polynomials.
  • The derived q-generating functions offer new insights into special function theory.
  • This work bridges classical and q-calculus through polynomial generating functions.