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An idea of Hermite is used to give a simple proof of Jacobi's generating function for Jacobi polynomials. One of the standard ways to prove the orthogonality of Legendre polynomials is to take their generating function (1 2xr + r 2 (1) n =O and show that the integral f( 2xr + r -2) l/2( 2xs + s2)_ 1/2 dx is a function of the variable rs. See, for example, Courant-Hilbert [1, pp. 85-86]. This proof was given by Legendre [6, p. 250]. Jacobi gave a generating function for a more general set of orthogonal polynomials Pn(a,i O(X). These polynomials can be defined by ( X)a(l + X) iP (a,f)(X) = ( I)n d n [(X)n+a(l + x)n+Y]. (2) n ~~2nn! dX~ [( It is easy to use (2) and integration by parts to prove fp(a,,8)(x)p (a,fi)(x)(l X)a(l + x) / dx = 0, m 7# n, a,/3 > 1. (3) n I Jacobi's generating function is 00 P p(a"f)(x)rn.= 2a+iR-l(I r + R)-a(, + r + R) a (4) n=O where R = (1 2xr + r2)1/2. His original proof used Lagrange's extension of Taylor's theorem. A second proof of this generating function was given a few years later by Tchebychef [9]. His proof was modeled on Legendre's proof mentioned above. He seems to have found this generating function independently from Jacobi, since he does not mention Jacobi's paper. Tchebychef's proof is a very complicated one which involves a number of changes of variables to reduce the integral Received by the editors September 9, 1977. AMS (MOS) subject classifications (1970). Primary 33A65; Secondary 42A56.

Jacobi’s generating function for Jacobi polynomials