I'm looking for a relationship similar to the one below, but the polynomial in question is a generalized Jacobi polynomial?

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Dudley J Benton

Chapter 6 of Abramowitz & Stegun, "Handbook of Mathematical Functions" You can get the PDF online free.

Mohamed Amine Bahayou

It will be nice to introduce some useful definitions (or a wiki link) to get feedback to your question. You can ask the authors of:

https://homepage.tudelft.nl/11r49/diff/

Article Differential equations for generalized Jacobi polynomials

Or put your question in https://math.stackexchange.com/

Good luck!

I have attached Chapter 16. The entire text is 78MB and too large but you can find it online for free. It's an excellent reference.