Some Properties of Generalized Gegenbauer Matrix Polynomials

Various new generalized forms of the Gegenbauer matrix polynomials are introduced using the integral representation method, which allows us to express them in terms of Hermite matrix polynomials. Certain properties for these new generalized Gegenbauer matrix polynomials such as recurrence relations and expansion in terms of Hermite matrix polynomials are derived. Further, several families of bilinear and bilateral generating matrix relations for these polynomials are established and their applications are presented. 1. Introduction Theory of generalized and multivariable special functions has provided new means of analysis to deal with the majority of problems in mathematical physics which find broad practical applications. Further, an extension to the matrix framework of special functions is special matrix functions. The study of special matrix polynomials is important due to their applications in certain areas of statistics, physics, and engineering. In recent years, some results in the theory of classical orthogonal polynomials have been extended to orthogonal matrix polynomials [1], which forms an emergent field and plays an important role from both the theoretical and practical point of view. Orthogonal matrix polynomials appear in connection with representation theory, matrix expansion problems, prediction theory, and in the reconstruction of matrix functions. The Laguerre and Hermite matrix polynomials and their extension and generalizations have been introduced and studied in [2–9] for matrices in whose eigenvalues are all situated in right open half-plane. If is the complex plane cut along the negative real axis and denotes the principal logarithm of , then represents . If is a matrix in with where (the spectrum of ) is the set of all the eigenvalues of , then denotes the image by of the matrix functional calculus acting on the matrix . Throughout this paper, we assume that is a positive stable matrix in ; that is, satisfies the following condition: First, we recall that the Chebyshev polynomials (CP) and Gegenbauer polynomials (GP) are defined in [10] as Next, we recall certain recently introduced Hermite matrix and Laguerre matrix polynomials. We mentioned these matrix polynomials in Table 1. Table 1: List of known Hermite matrix and Laguerre matrix polynomials. Due to the importance of generalized Hermite matrix polynomials, which find broad practical applications recently, Batahan [2] introduces a matrix version of Chebyshev polynomials in terms of 2VHMaP (Table 1(I)). To give an idea of the procedure adopted in [11], we use 2VHMaP to