New Generating Functions for Chebyshev, Gegenbauer, and Jacobi Matrix Polynomials

Generating functions are a fundamental tool in the analysis of orthogonal polynomials, but their extension to the matrix setting has remained fragmented across different polynomial families. In a new article published in Mediterranean Journal of Mathematics, Emilio Defez and Michael M. Tung present a unified approach to deriving generating functions for the matrix analogues of the Chebyshev, Gegenbauer, and Jacobi polynomials.

The work does not stop at the matrix level. Alongside the matrix generating functions, the authors establish several explicit identities for the corresponding scalar polynomial families. These results provide new analytical tools for a class of polynomials that play a growing role in the numerical solution of matrix differential equations, the approximation of inverse Laplace transforms, and the high-accuracy computation of matrix functions such as the exponential.

By bringing three classical families under a single derivation framework, the article offers a compact reference for researchers working with matrix orthogonal polynomials and opens the door to analogous results for other polynomial sequences.

Publication Details

  • Title: New Generating Functions for Chebyshev, Gegenbauer, and Jacobi Matrix Polynomials
  • Authors: E. Defez, M. M. Tung
  • Journal: Mediterranean Journal of Mathematics, Vol. 23, 2026, Article 118
  • DOI: 10.1007/s00009-026-03119-x

Main Contributions

  • A unified derivation of generating functions for Chebyshev, Gegenbauer, and Jacobi matrix polynomials.
  • New explicit identities for the corresponding scalar polynomial families.
  • A consistent analytical framework that connects three classical polynomial sequences in the matrix setting.

Why This Matters

Matrix orthogonal polynomials are increasingly used in numerical linear algebra and in the approximation of matrix functions. Having a unified source of generating functions simplifies the derivation of new series expansions and can reduce the analytical effort required when adapting classical polynomial techniques to the matrix case.