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 ( α , β ) , generated from the classical second-order Jacobi differential expression ℓ α , β , k [ y ] ( t ) = 1 w α , β ( t ) ( ( - ( 1 - t ) α + 1 ( 1 + t ) β + 1 y ′ ( t ) ) ′ + k ( 1 - t ) α ( 1 + t ) β y ( t ) ) ( t ∈ ( - 1 , 1 ) ) , in the Hilbert space L α , β 2 ( - 1 , 1 ) ≔ L 2 ( ( - 1 , 1 ) ; w α , β ( t ) ) , where w α , β ( t ) = ( 1 - t ) α ( 1 + t ) β , that has the Jacobi polynomials { P m ( α , β ) } m = 0 ∞ as eigenfunctions; here, α , β > - 1 and k is a fixed, non-negative constant. More specifically, for each n ∈ N , we explicitly determine the unique left-definite Hilbert–Sobolev space W n , k ( α , β ) ( - 1 , 1 ) and the corresponding unique left-definite self-adjoint operator B n , k ( α , β ) in W n , k ( α , β ) ( - 1 , 1 ) associated with the pair ( L α , β 2 ( - 1 , 1 ) , A k ( α , β ) ) . The Jacobi polynomials { P m ( α , β ) } m = 0 ∞ form a complete orthogonal set in each left-definite space W n , k ( α , β ) ( - 1 , 1 ) and are the eigenfunctions of each B n , k ( α , β ) . Moreover, in this paper, we explicitly determine the domain of each B n , k ( α , β ) as well as each integral power of A k ( α , β ) . The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓ α , 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 A k ( α , β ) .