While I am reading "On some dual integral equations occurring in potential problems with axial symmetry" by C. J. Tranter, Quarterly Journal of Mechanics & Applied Math (1950) p. 414, the author stated that link to the article $$ \mathcal{F}_m (\alpha,\gamma,x) = {}_2F_1 (-m,\alpha+m;\gamma;x) \, , $$ where $\mathcal{F}$ is Jacobi polynomial. From Wikipedia, I have found another definition which reads $$ \mathcal{F}_m (\alpha,\gamma,x) = \frac{(\alpha+1)_m}{m!} {}_2F_1 (-m, 1+\alpha+\gamma+m;\alpha+1;\tfrac{1}{2} (1-x))\, , $$ with $(\alpha+1)_m$ being the Pochhammer's symbol. Apparently the two definitions are not equivalent.

I was wondering whether other definitions of Jacobi polynomials have also been introduced in the literature. Any clarification is highly appreciated.




Here are some references:


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