I am told that $$P_n(x) ={}_2F_1\left(-n,n+1;1;\tfrac{1-x}{2}\right),$$ where $P_n(x)$ is Legendre polynomial and ${}_2F_1\left(a,b;c;z\right)$ is hypergeometric function. I am just wondering how to prove it.
2026-03-25 20:36:32.1774470992
Show that $P_n(x) ={}_2F_1\left(-n,n+1;1;\frac{1-x}{2}\right)$.
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You can expand $P_n(x)$ in powers of $1-x$ starting with Rodrigues's formula and compare the result with the hypergeometric series.