On the Rodrigues formula solution of the hypergeometric-type differential equation

Abstract

In this paper, we present a new systematic approach to the solution of the hypergeometric-like differential equation and its associated equation. The method produces, tout court, the general solution of these equations in the form of a combination of a standard Rodrigues formula and a 'generalized' Rodrigues formula, of a type due originally to Gonçalves [5] and recently considered, again, by Area et al [1]. In addition, a novel analysis of a class of integrals determining the generalized Rodrigues formulae is given, which complements an original analysis of Area et al [1]. Finally, the relation between the hypergeometric-type differential equation and the hypergeometric equation is elucidated further, following the work of Koepf and Masjed-Jamel [7]. Mathematics Subject Classification: 33C25; 33C45

Non-hermitian Dirac Hamiltonian in the presence of local Fermi velocity

“…After fixing these functions we return to (A.8) to calculate y (z) and λ [30,36,37]. Here we use a property of hypergeometric functions, to evaluate λ, which tells that all higher order derivative of a hypergeometric function is also a hypergeometric function.…”

Generalizations of Rodrigues type formulas for hypergeometric difference equations on nonuniform lattices

“…Recently, inspired by [3], W. Robin gave a more general Rodrigues formula [24] y n (x) = 1 ρ(x) d n dx n ρ(x)σ n (x)…”

“…A. Wilson, M. Ismail [14,15,16,17]; F. Nikiforov, K. Suslov, B. Uvarov, N. M. Atakishiyev [8,9,10,18,19,20]; G. George, M. Rahman [21]; T. H. Koornwinder [22]; and many other researchers like R. Álvarez-Nodarse, K. L. Cardoso, I. Area, E. Godoy, A. Ronveaux, A. Zarzo, W. Robin, T. Dreyfus, V. Kac, P. Cheung, and L. K. Jia, J. F. Cheng, Z.S, Feng [23][24][25][26][27][28][29][30][31][32].…”

Explanation of Nikiforov-Uvarov method in quantum mechanics – with a view toward algebraic geometry

“…At present, y(s) and λ are still unknowns in the sense of eigenvalue problem. According to the theory of hypergeometric function [6], the relevant solutions to Eq. (11)(indexed by integers n = 0, 1, 2, ...) are given by…”