J. Koekoek

TL;DR: In this paper, it was shown that the generalized Laguerre polynomials (a+ for N > 0) satisfy a unique differential equation of the form 00 N ai (x)y(1)(x) + xy "(x)+ ( + 1-x)-y'(x)+ ny(x)) = 0, 1=0 where {a,(x)}1o are continuous functions on the real line and {a'(ex)1i are independent of the degree n.

TL;DR: In this article, the generalized Jacobi polynomials of the form M ∑ i=0 ∞ a i (x)y (i) (x)+N ∑ ∞ b i y (y)(i)(x) +(1−x 2 )y″(x)+[β−α−(α+β+2)x]y′(x)-n(n+α +β+1)y

TL;DR: In this article, the authors obtained all spectral type differential equations satisfied by the Sobolev-type Laguerre polynomials, which generalizes the results found in 1990 by the first and second author in the case of the generalized Laguers.

TL;DR: In this article, the q-derivative operator Dq is defined by [formula] for functions which are differentiable at x = 0, and we have for every positive integer n[formula], for every function ǫ whose nth derivative at x=0 exists.

TL;DR: In this paper, the generalized Jacobi polynomials were used to find explicit formulas for the coefficients of these differential equations, which is a consequence of the Jacobi inversion formula which is proved in this paper.