Several properties of Racah polynomials

In this paper, bilinear and bilateral generating functions for Racah polynomials are derived, along with a theorem that provides a systematic approach for obtaining these functions. Furthermore, a new recurrence relation and an integral representation for Racah polynomials are established, enhancing their analytical framework. Special attention is given to the limiting cases of Racah polynomials, including Hahn, dual Hahn and Meixner polynomials, for which new recurrence relations are obtained. In particular, a novel integral representation for the dual Hahn polynomial is introduced, offering additional insights into its structural properties. These results contribute to the broader understanding of orthogonal polynomials, enhancing their theoretical significance and potential applications. By expanding the known properties of these polynomials, the findings may provide a basis for further mathematical research and applications in areas such as combinatorics, mathematical physics, and special functions.