Legendre polynomials, Legendre–Stirling numbers, and the left-definite spectral analysis of the Legendre differential expression

Abstract

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A( k) in L 2(−1,1), generated from the classical second-order Legendre differential equation ℓ L,k [y](t)=−((1−t 2)y′)′+ky=λy (t∈(−1,1)), that has the Legendre polynomials { P m ( t)} m=0 ∞ as eigenfunctions; here, k is a fixed, nonnegative constant. More specifically, for k>0, we explicitly determine the unique left-definite Hilbert–Sobolev space W n ( k) and its associated inner product (·,·) n, k for each n∈ N. Moreover, for each n∈ N, we determine the corresponding unique left-definite self-adjoint operator A n ( k) in W n ( k) and characterize its domain in terms of another left-definite space. The key to determining these spaces and inner products is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓ L, k [·]. In turn, the key to determining these powers is a remarkable new identity involving a double sequence of numbers which we call Legendre–Stirling numbers.