On some differential transformations of hypergeometric equations

30th International Colloquium on Group Theoretical Methods in Physics

Hounkonnou, M.N.^1 ; Ronveaux, A.^1,2

International Chr. in Mathematical Physics and Applications (ICMPA-UNESCO Chr.), University of Abomey-Calavi, 072 B.P., 50 Cotonou, Benin^1

Departement de Mathematiques, Catholic University of Louvain, Chemin du Cyclotron,2, Louvain-la-Neuve

B-1348, Belgium^2

关键词: Algebraic transformations;    Differential transformation;    Exceptional polynomials;    Hermite polynomials;    Heun equations;    Hypergeometric;    Laguerre;    Polynomial functions;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012044/pdf
DOI  :  10.1088/1742-6596/597/1/012044

来源: IOP

【 摘 要 】

Many algebraic transformations of the hypergeometric equation σ(x)z"(x) + τ(x)z'(x) + lz(x) = 0, where σ, τ, l are polynomial functions of degrees 2 (at most), 1, 0, respectively, are well known. Some of them involve x = x(t), a polynomial of degree r, in order to recover the Heun equation, extension of the hypergeometric equation by one more singularity. The case r = 2 was investigated by K. Kuiken (see 1979 SIAM J. Math. Anal. 10 (3) 655-657) and extended to r = 3,4, 5 by R. S. Maier (see 2005 J. Differ. Equat. 213 171 - 203). The transformations engendered by the function y(x) = A(x)z(x), also very popular in mathematics and physics, are used to get from the hypergeometric equation, for instance, the Schroedinger equation with appropriate potentials, as well as Heun and confluent Heun equations. This work addresses a generalization of Kimura's approach proposed in 1971, based on differential transformations of the hypergeometric equations involving y(x) = A(x)z(x) + B(x)z'(x). Appropriate choices of A(x) and B(x) permit to retrieve the Heun equations as well as equations for some exceptional polynomials. New relations are obtained for Laguerre and Hermite polynomials.