# Difference equation of hypergeometric type

From specialfunctionswiki

A second order difference equation of hypergeometric type is of the form $$\sigma(s)\nabla \Delta y(s) + \tau(s)\Delta y(s)+\lambda y(s)=0,$$ where $\sigma(s)$ is a polynomial of degree at most $2$, $\tau(s)$ is a polynomial of degree at most $1$, and $\lambda \in \mathbb{R}$.

# Properties

**Theorem:** The difference equation of hypergeometric type can be written in the self-adjoint form
$$\Delta [\sigma(s)\rho(s)\nabla y(s)]+\lambda \rho(s)y(s)=0,$$
where the function $\rho$ satisfies a Pearson difference equation
$$\Delta [ \sigma(s)\rho(s) ] = \tau(s)\rho(s).$$

**Proof:** █