Nonlinear characterizations for functions of hypergeometric type and their derivatives of any order (Q1331001)

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.