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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
Summary
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.
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.
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