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

  • Authors:

  • W. N. Everitt;K. H. Kwon;L. L. Littlejohn;R. Wellman;G. J. Yoon

  • Affiliations:

  • Department of Mathematics, University of Birmingham, Birmingham, B15 2TT, England, UK;Division of Applied Mathematics, Kaist, 373-1 Kusong-Dong, Yusong-Ku, Daejeon 305-701, Republic of Korea;Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA;Department of Mathematics, Westminster College, Salt Lake City, UT 84105, USA;School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-Dong, Dongdaemun-Gu, Seoul 130-722, Republic of Korea

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2007

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Abstract

We develop the left-definite analysis associated with the self-adjoint Jacobi operator , generated from the classical second-order Jacobi differential expression ,in the Hilbert space , where wα,β(t)=(1-t)α(1+ t)β, that has the Jacobi polynomials as eigenfunctions; here, α,β-1 and k is a fixed, non-negative constant. More specifically, for each , we explicitly determine the unique left-definite Hilbert-Sobolev space and the corresponding unique left-definite self-adjoint operator in associated with the pair . The Jacobi polynomials form a complete orthogonal set in each left-definite space and are the eigenfunctions of each . Moreover, in this paper, we explicitly determine the domain of each as well as each integral power of . The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓα,β,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 .