Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case

Abstract

Motivated by the G.H. Hardy's 1939 results [G.H. Hardy, Notes on special systems of orthogonal functions II: On functions orthogonal with respect to their own zeros, J. London Math. Soc. 14 (1939) 37–44] on functions orthogonal with respect to their real zeros λn, n=1,2,... , we will consider, under the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval (0,1), that is, the functions f(z)=z^ν F(z), ν in R, where F is entire and,

$\int_0 1 f(λ_n t)f(λ_m t)t (1-t) dt=0, α>-1-2ν, β>-1

when n≠m. Considering all possible functions on this class we obtain a new family of generalized Bessel functions including Bessel and hyperbessel functions as special cases.