recurrence relations of Jacobi polynomials

Question

You'll commonly find recurrence relations of Jacobi polynomials in terms of $n$, $\alpha$, or $\beta$, e.g., $$ \begin{align*} P_n^{(\alpha, \beta)}(x) &= c_0(x) P_{n-1}^{(\alpha, \beta)}(x) + c_1(x) P_{n-2}^{(\alpha, \beta)}(x),\\ P_n^{(\alpha, \beta)}(x) &= c_2(x) P_n^{(\alpha-1, \beta)}(x) + c_3(x) P_{n}^{(\alpha-2, \beta)}(x),\\ P_n^{(\alpha, \beta)}(x) &= c_4(x) P_n^{(\alpha, \beta-1)}(x) + c_5(x) P_{n}^{(\alpha, \beta-2)}(x). \end{align*} $$ I'm wondering if there are relations "around the corner" à la $$ P_n^{(\alpha, \beta)}, P_{n+1}^{(\alpha, \beta)}, P_{n}^{(\alpha+1, \beta)}. $$ Any hints?

Answer 1

On https://arxiv.org/abs/1509.08213 I found $$ 0 = (2n + \alpha + \beta)P^{(\alpha−1,\beta)}_n(x) - (n + \alpha + \beta)P^{(\alpha,\beta)}_n(x) + (n + \beta)P^{(\alpha,\beta)}_{n-1}(x), $$ $$ 0 = (2n + \alpha + \beta)P^{(\alpha,\beta−1)}_n(x) - (n + \alpha + \beta)P^{(\alpha,\beta)}_n(x) - (n + \alpha)P^{(\alpha,\beta)}_{n−1}(x). $$