On A Matrix Hypergeometric Differential Equation

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...

The Generalized Matrix Valued Hypergeometric Equation

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 hypergeometri...

Quasimodular solutions of a differential equation of hypergeometric type

Let p ≥ 5 be a prime number and Fp−1(τ) be the solution of the above differential equation for k = p−1 which is modular on SL2(Z) (such a solution exists and is unique up to a scalar multiple). For any zero τ0 in H of the form Fp−1(τ), the value of the jfunction at τ0 is algebraic and its reduction modulo (an extension of) p is a supersingular j-invariant of characteristic p, and conversely, al...

Differential Equation of Sensitivity Matrix for Final Velocity Constraint

The Hyperbolic Schwarz Map for the Hypergeometric Differential Equation

The Schwarz map of the hypergeometric differential equation is studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is the hyperbolic 3-space. This map can be considered to be a lifting to the 3-space of the Schwarz map. This paper studies the singularities of this map, and visual...