Simplify Jacobi Polynomials

Question

How can I force Mathematica to use the identities satisfied by Jacobi polynomials $$ (1-\cdot)P_n^{(\alpha+1,\beta)} = \frac{2}{2n+\alpha+\beta+2}\left((n+\alpha+1)P_n^{(\alpha,\beta)}-(n+1)P_{n+1}^{(\alpha,\beta)}\right) $$ $$ (1+\cdot)P_n^{(\alpha,\beta+1)} = \frac{2}{2n+\alpha+\beta+2}\left((n+\beta+1)P_n^{(\alpha,\beta)}+(n+1)P_{n+1}^{(\alpha,\beta)}\right) $$ to simplify expressions involving this polynomials in the sense that it is more preferable to have many Jacobi Polynomials multiplied by constants than a few multiplied by explicit monomials.

By using FullSimplify with Assumptions to tell Mathematica that both $\alpha$ and $\beta$ are greater than $-1$ I still obtain expressions containing products of Jacobi polynomials with $(1-t)$ or $(1+t)$

Simple example

When I input

FullSimplify[(1 - x) JacobiP[n, \[Alpha]+1, \[Beta], x], Assumptions ->   n \[Element] Integers && n > 0 && \[Alpha] > -1 && \[Beta] > -1 && x \[Element] Reals]

Mathematica just returns

-(-1 + x) JacobiP[n, \[Alpha]+1, \[Beta], x]

whereas I would like to get

2/(2 n + \[Alpha] + \[Beta] + 2) ((n + \[Alpha] + 1) JacobiP[n, \[Alpha], \[Beta], x] - (n + 1) JacobiP[n + 1, \[Alpha], \[Beta], x])

Answer 1

If you want to put an expression into a specific form, particularly if the desired form is not simpler in the normal sense (i.e., using the default complexity function), it may be easier using replacement rules along with ReplaceAll

Clear["Global`*"]

The given identities are equivalent to these rules.

repl = {(1 - t_)*JacobiP[n_, α_, β_, t_] :> 
    2/(2 n + α + β + 1)*
     ((n + α)*JacobiP[n, α - 1, β, t] -
       (n + 1)*JacobiP[n + 1, α - 1, β, t]),
   (1 + t_)*JacobiP[n_, α_, β_, t_] :> 
    2/(2 n + α + β + 1)*
     ((n + β)*JacobiP[n, α, β - 1, t] +
       (n + 1)*JacobiP[n + 1, α, β - 1, t])};

expr1 = (1 - x) JacobiP[n, α, β, x];

expr2 = expr1 /. repl

(* (1/(1 + 2 n + α + β))2 ((n + α) JacobiP[
     n, -1 + α, β, x] - (1 + n) JacobiP[
     1 + n, -1 + α, β, x]) *)

Verifying,

expr1 == expr2 // FullSimplify

(* True *)

Similarly,

expr3 = (1 + x) JacobiP[n, α, β, x];

expr4 = expr3 /. repl

(* (1/(1 + 2 n + α + β))2 ((n + β) JacobiP[
     n, α, -1 + β, x] + (1 + n) JacobiP[
     1 + n, α, -1 + β, x]) *)

Verifying,

expr3 == expr4 // FullSimplify

(* True *)