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C~p;81 , 82 . . . (~p) n n=O n! (Bq)n " Burchnall's generating function (6) is essentially (7) with p = q = 0, c = a-l, x = zb -I, and t replaced by bt/2. ,~p; 81 .... ,Sq;-X); then (l-t) -c p+2Fq(C/2,(c+l)/2,~l ..... ~p;B 1 ..... 8q; 4xt(l-t) -2) = ~ On(X)(C)ntn/n~ • n=0 For p = q = 0, On(X) = 2F0(-n,c+n;-;-x) = Yn(bX,c+l,b), so that (8) is the generating function for the yn[Z;a,b), with z = bx, a = c+l, and the same b. 5. A. 4) for integral (i0') a Z tnyn(Z;a-n-l,b). n=0 (i0) may be written also as (l-t)-ly2_a(Z/(l-t-l);a,b) -~ [ tnyn(Z;a-n-l,b ).
Interest• (f,g) = (g,f), the inner product now satisfies the Much of the classical theory goes through in this broader For the case in point, however, a different kind of orthogonality is of It occurs already in  and has been studied in detail by John W. Jayne  who calls it the Bessel alternative. D. M. Krall in fact a distribution. Still more general points of view are  start from a weight "function" that is, In the case of the classical orthonormal polynomials (Jacobi, Laguerre, Hermite), use of the Fourier transform permits to recover the usual weight functions from the distributional successful in the case of the BP.
3... (2r-l) [ (-2~-l)(-z)m m=O [ (-l)m(-2~ -I) zm+r = m=0 ~ m=0 (2r÷l) (2r+2) m! 3...