Numerical evidence indicates that Jacobi polynomials with negative integer parameters satisfy the identity $$P_n^{(-m,-k)}(x)=\left(\frac{x-1}{2}\right)^m\left(\frac{1+x}{2}\right)^kP_{n-m-k}^{(m,k)}(x),$$ where $n\ge m+k$. How this identity can be proved?

I came across this supposed identity when comparing some linearisation relations for the product of two Laguerre polynomials from https://cds.cern.ch/record/1378854?ln=en (after correcting typos) and https://iopscience.iop.org/article/10.1088/0305-4470/18/9/022 but such a proof (assuming both papers are correct) is too complicated and I prefer a direct proof which then I can use to argue that the results of these two papers are equivalent to each other.

**P.S.** The identity can be proved also by using
$$P_n^{(-m,\beta)}(x)=\frac{\Gamma(n+\beta+1)}{\Gamma(n+\beta+1-m)}\frac{(n-m)!}{n!}\left(\frac{x-1}{2}\right)^m\,P_{n-m}^{(m,\beta)}(x),$$
which can be found in the book G. Szego, Orthogonal Polynomials (formula (4.22.2)), in combination with the symmetry relation $$P_n^{(\alpha,\beta)}(x)=(-1)^n\,P_n^{(\beta,\alpha)}(-x).$$