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

We develop the left-definite analysis associated with the self-adjoint Jacobi operator A , ) k , generated from the classical secondorder Jacobi differential expression , ,k[y](t) = 1 w , (t) ((−(1 − t) +1(1 + t) +1y′(t))′ + k(1 − t) (1 + t) y(t)) (t ∈ (−1, 1)), in the Hilbert space L2 , (−1, 1) := L2((−1, 1); w , (t)), where w , (t) = (1 − t) (1 + t) , that has the Jacobi polynomials {P ( , ) m }∞m=0 as eigenfunctions; here, , > − 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 (−1, 1) and the corresponding unique left-definite selfadjoint operator B , ) n,k in W , ) n,k (−1, 1) associated with the pair (L2 , (−1, 1), A , ) k ). The Jacobi polynomials {P ( , ) m }∞m=0 form a complete orthogonal set in each left-definite space W , ) n,k (−1, 1) and are the eigenfunctions of each B , ) n,k . Moreover, in this paper, we explicitly determine the domain of each B , ) n,k as well as each integral power of A , ) k . The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of , ,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 . © 2006 Published by Elsevier B.V. MSC: primary 33C65; 34B30; 47B25; secondary 34B20; 47B65

[1]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[2]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[3]  Lance L. Littlejohn,et al.  The left-definite spectral theory for the classical Hermite differential equation , 2000 .

[4]  P. Goldbart,et al.  Linear differential operators , 1967 .

[5]  G. Szegő Zeros of orthogonal polynomials , 1939 .

[6]  A. Zettl POWERS OF REAL SYMMETRIC DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS , 1976 .

[7]  H. Weyl,et al.  Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen , 1910 .

[8]  W. N. Everitt,et al.  PRODUCTS OF DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS , 1978 .

[9]  F. W. Schäfke,et al.  S -Hermitesche Rand-Eigenwertprobleme , 1965 .

[10]  Steven Roman The Umbral Calculus , 1984 .

[11]  P. B. Bailey THE SLEIGN2 STURM-LIOUVILLE CODE , 2001 .

[12]  Wn Everitt,et al.  Legendre polynomials and singular differential operators , 1980 .

[13]  Yoshimi Saito,et al.  Eigenfunction Expansions Associated with Second-order Differential Equations for Hilbert Space-valued Functions , 1971 .

[14]  Leon M. Hall,et al.  Special Functions , 1998 .

[15]  L. Comtet,et al.  Advanced Combinatorics: The Art of Finite and Infinite Expansions , 1974 .

[16]  Lance L. Littlejohn,et al.  On Properties of the Legendre Differential Expression , 2002 .

[17]  W. Rudin Real and complex analysis , 1968 .

[18]  Lance L. Littlejohn Symmetry Factors for Differential Equations , 1983 .

[19]  Lance L. Littlejohn,et al.  A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations , 2002 .

[20]  M. Katětov Linear operators. II , 1951 .

[21]  Francisco Marcellán,et al.  On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials , 2002 .

[22]  Rolf Vonhoff,et al.  A Left-Definite Study of Legendre’s Differential Equation and of the Fourth-Order Legendre Type Differential Equation , 2000 .

[23]  A. Krall A review of orthogonal polynomials satisfying boundary value problems , 1988 .

[24]  Lance L. Littlejohn,et al.  Legendre polynomials, Legendre--Stirling numbers, and the left-definite spectral analysis of the Legendre differential expression , 2002 .