Nonlinear characterizations for functions of hypergeometric type and their derivatives of any order
The so-called functions of hypergeometric type \(y\equiv y_ \nu(x)\) are solutions of the differential equation \(\sigma (z)y''+ \tau(z) y'+ \lambda y=0\), where \(\sigma\), \(\tau\) and \(\lambda\) are polynomials of degrees, at most 2, 1 and 0, respectively, and \(\nu\) is a root of the equation \(\lambda+ \nu\tau'+ {1\over 2} \nu(\nu -1) \sigma''=0\). The authors establish two interesting characterizations for these functions and their derivatives. Each characterization consists of four different quadratic differential-difference equations among \(y_ \nu^{(k)} (z)\). Applications to the hypergeometric type polynomials are shown. In particular, the coefficients of the relationships are explicitly given for the classical orthogonal polynomials of Jacobi, Laguerre, Hermite and Bessel.
scientific article; zbMATH DE number 617439