On a differential equation for Koornwinder’s generalized Laguerre polynomials

Abstract:

Koornwinder’s generalized Laguerre polynomials $\left \{ {L_n^{\alpha ,N}(x)} \right \}_{n = 0}^\infty$ are orthogonal on the interval $[0,\infty )$ with respect to the weight function $\frac {1}{{\Gamma (\alpha + 1)}}{x^\alpha }{e^{ - x}} + N\delta (x),\alpha > - 1,N \geq 0$. We show that these polynomials for $N > 0$ satisfy a unique differential equation of the form \[ N\sum \limits _{i = 0}^\infty {{a_i}(x){y^{(i)}}(x) + xy''(x) + (\alpha + 1 - x)y’(x) + ny(x)} = 0,\] where $\left \{ {{a_i}(x)} \right \}_{i = 0}^\infty$ are continuous functions on the real line and $\left \{ {{a_i}(x)} \right \}_{i = 1}^\infty$ are independent of the degree $n$. If $N > 0$, only in the case of nonnegative integer values of $\alpha$ this differential equation is of finite order.