Racah polynomials

Remarks

Since $(-x)_k (x + \gamma + \delta + 1)_k = \prod_{j=0}^{k-1} \left( j(j + \gamma + \delta + 1) - \lambda(x) \right)$, each $R_n$ is a polynomial of degree $n$ in $\lambda(x)$. Example: $R_1 = 1 + \frac{(\alpha + \beta + 2) \, \lambda(x)}{(\alpha + 1)(\beta + \delta + 1)(\gamma + 1)}$. They are orthogonal on $x = 0, 1, \dots, N$ for the weight $$w(x) = \frac{(\alpha + 1)_x (\beta + \delta + 1)_x (\gamma + 1)_x (\gamma + \delta + 1)_x \left( \frac{\gamma + \delta + 3}{2} \right)_x}{(-\alpha + \gamma + \delta + 1)_x (-\beta + \gamma + 1)_x \left( \frac{\gamma + \delta + 1}{2} \right)_x (\delta + 1)_x \, x!},$$ that is, $\sum_{x=0}^{N} w(x) R_m(\lambda(x)) R_n(\lambda(x)) = 0$ for $m \ne n$; the weight is positive for suitable ranges of the parameters. The balanced condition holds: the sum of the lower parameters exceeds the sum of the upper ones by $1$. Non-example: with $\alpha + 1 = -N$, the degree $n = N + 1$ is not allowed, since the term $k = N + 1$ has the factor $(\alpha + 1)_{N+1} = (-N)_{N+1} = 0$ in its denominator.