On a generalization of the Rogers generating function

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.