Jacobi-Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression

Abstract

We develop the left-definite analysis associated with the self-adjoint Jacobi operator A(k)((alpha,beta)), generated from the classical second order Jacobi differential expression l(alpha,beta,k)[y](t) = 1/w(alpha,beta(t)) ((-(1-t)(alpha+1) (1+t)(beta+1) y ' (t))' + k(1-t)(alpha)(1+t)(beta) y(t)) (t is an element of (-1, 1). in the Hilbert space L-alpha,beta(2)(-1,1) := L-2(-1, 1); w(alpha,beta)(t)), where w(alpha,beta)(t) = (1-t)alpha(1+t)beta , that has the Jacobi polynomials {P-m((alpha,beta))}(m=0)(infinity) as eigenfunctions; here, alpha,beta > -1 and k is a fixed, non-negative constant. More specifically, for each n is an element of N, we explicitly determine the unique left-definite Hilbert-Sobolev space W-n,k((alpha,beta))(-1,1) and the corresponding unique left-definite self-,k adjoint operator B-n,k((alpha,beta))(-1, 1) associated with the pair (L-alpha,beta(2)(-1, 1) A(k)((alpha,beta))). The Jacobi polynomials (P-m((alpha,beta))}(m=0)(infinity) form a complete orthogonal set in each left-definite space W-n,k((alpha,beta))(- 1, 1) and are the eigenfunctions of each B-n,k((alpha,beta)). Moreover, in this paper, we explicitly determine the domain of each B-n,k((alpha,)beta) as well as each intergral power of A(k)((alpha,beta)). The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of l(alpha,beta,k)[.]. In turn, the key to determining these powers is a double sequence of numbers which we introduce in this paper as the Jacobi-Stirling numbers. Some properties of these numbers, which in some ways behave like the classical Stirling numbers of the second kind, are established including a remarkable, and yet somewhat mysterious, identity involving these numbers and the eigenvalues of A(k)((alpha,beta)) (c) 2006 Published by Elsevier B.V.