The Generalized Matrix Valued Hypergeometric Equation
نویسنده
چکیده
Abstract. The matrix valued analog of the Euler’s hypergeometric differential equation was introduced by Tirao in [1]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao’s equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [1] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, this functions are analytic for |z| < 1 and we give a necesary condition for the convergence on the unit circle |z| = 1.
منابع مشابه
The matrix-valued hypergeometric equation.
The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1, 11-13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123-162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39-83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127-172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1-24]. The hypergeometric function kno...
متن کاملNoncommutative Hypergeometric and Basic Hypergeometric Equations
Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138–8141] considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tirao’s result, extended to the more general setting of hypergeometr...
متن کاملA Hypergeometric Function Transform and Matrix-valued Orthogonal Polynomials
The spectral decomposition for an explicit second-order differential operator T is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomial...
متن کاملNoncommutative Extensions of Ramanujan’s 1ψ1 Summation ∗
Using functional equations, we derive noncommutative extensions of Ramanujan's 1 ψ 1 summation. 1. Introduction. Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have been the subject of recent study, see e.g. the papers by Duval and Ovsienko [DO], Grünbaum [G], Tirao [T], and some of the references mentioned therein. Of course, t...
متن کاملUniversity of Cambridge Integro-diierential Equations and Generalized Hypergeometric Functions Integro-diierential Equations and Generalized Hypergeometric Functions Are Neither Zero nor a Negative In- Teger, and 2 0; 2) There Exist Smooth Functions and Such That the Generalized Hypergeometric Function
This paper is concerned with the integro-diierential equations y 0 (t) = ay(t) + Z 1 0 y(qt) d(q) + Z 1 0 y 0 (qt) d(q); t 0 and y(t) + Z 1 0 y(qt) d(q) + Z 1 0 y 0 (qt) d(q) = 0; t 0; where a is a complex constant, while and are complex-valued functions of bounded variation on 0; 1]. The main motivation for the study of these two equations is that for every integers B + 1 A 0 and real constant...
متن کامل