gegenbauer polynomial

Question

Usually, Gegenbuaer polynomial is denoted by $C^{(\lambda )}_{n}(x)$ with $\lambda >-1/2$. My question: is it possible to generalize Gegenbuaer polynomial for $Re(\lambda)>-1/2, \lambda \in \mathbb{C}$? Is there any reference for this problem? Thank you.

Answer 1

The Gegenbauer polynomials can be defined as $$ C_n^{(\lambda)} = \frac{(2\lambda)_n}{n!} {}_2F_1\left(-n, n+2\lambda; \lambda + \frac{1}{2}; \frac{(1-x)}{2}\right), $$ where ${}_2F_1$ is the hypergeometric function. As long as $\lambda \neq -\frac{1}{2}, -1, -\frac{3}{2}, -2, -\frac{5}{2}, \ldots$ (which you assume with $Re(\lambda) > -\frac{1}{2}$ this hypergeometric function is well defined.

As far as I can check all common relations for the Gegenbauer polynomials generalize for $Re(\lambda) > -\frac{1}{2}$. This means that the orthogonality relations, generating function, solution to the differential equation, Rodrigues formula and three-term recurrence relations extend to $Re(\lambda) > -\frac{1}{2}$.

A good reference is Classical and quantum orthogonal polynomials in one variable from Mourad Ismail, pages 94-98.