An Addition Theorem for Hahn Polynomials: The Spherical Functions

An addition formula for the Hahn polynomials $Q_k (x;\alpha ,\beta ,N)$ is derived for the parameter values $\beta = - N - 1$, $\alpha \ne - 1, - 2, \cdots , - N$, $N = 1,2,3, \cdots $. The method is to realize $Q_k $ as a spherical function for the values $\alpha = - N - 1, - N - 2, \cdots $ and to use harmonic analysis on the finite homogeneous space $(S_b \times S_a )\backslash S_{a + b} $ where $b = N$, $a = - \alpha - 1$ and $S_n $ is the symmetric group on n objects $(n = 1,2, \cdots )$.