Szegö on Jacobi Polynomials

One of the interesting features in the development of analysis in the twentieth century is the remarkable growth, in various directions, of the theory of orthogonal functions. Two brilliant achievements on the threshold of this century—Fejér's paper on Fourier series and Fredholm's papers on integral equations—have been acting as a powerful inspiring source of attraction, inviting analysts to delve deeper into the theory of orthogonal functions and their applications. First come, due to their simplicity, the trigonometric functions {sin mx, cos mx} which serve as a yardstick for orthogonal functions in general. Next we may consider orthogonal polynomials, of which Jacobi polynomials are a special case. Let us recall the general definition of orthogonal polynomials. A weightfunction p(x), non-negative in a given interval (a, b), finite or infinite, and such that all "moments" fap(x)x dx = ar exist, (r—0, 1, 2, • • • ), with «o>0 , gives rise to a unique system of orthogonal and normal polynomials n(x) = anx n