We prove three theorems on Wilson functions (these special functions were introduced by Groenevelt in 2003). These theorems were stated without proof and applied for the proof of a summation formula related to automorphic forms in our paper .
We prove that a relatively general even function f(x) (satisfying a vanishing condition, and also some analyticity and growth conditions) on the real line can be expanded in terms of a certain function series closely related to the Wilson functions introduced by Groenevelt in 2003. The coefficients in the expansion of f will be inner products in a suitable Hilbert space of f and some polynomials closely related to Wilson polynomials (these are well-known hypergeometric orthogonal polynomials).
on the domain [−1,1] t/ Uk(x). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in . Various
bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof
of the uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms in .