Fourier-Jacobi Transform of Analytic Functions which are (Almost) Periodic in the Imaginary Direction

摘要

The authors show that the Fourier-Jacobi transform of index (alpha, beta), alpha > -1, beta is a member of R, maps functions of the form t-> phi (1-2 tanh squared t) cosh (sup minus nu) t, with phi an entire analytic function, bijectively onto the functions x -> Gamma(1/2(nu-alpha-beta-1+ix)) Gamma(1/2(nu-alpha-beta-1-ix)) Psi(x). Here Psi is an even and entire analytic function of sub-exponential growth. The treatment is based on recurrence relations.