Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
The left-definite spectral theory for the classical Hermite differential equation
Journal of Computational and Applied Mathematics - Special issue on numerical analysis in the 20th century vol. 1: approximation theory
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Journal of Computational and Applied Mathematics
Left-definite theory with applications to orthogonal polynomials
Journal of Computational and Applied Mathematics
Legendre-Stirling permutations
European Journal of Combinatorics
Hilbert scales and Sobolev spaces defined by associated Legendre functions
Journal of Computational and Applied Mathematics
Characterization of self-adjoint ordinary differential operators
Mathematical and Computer Modelling: An International Journal
Jacobi-Stirling polynomials and P-partitions
European Journal of Combinatorics
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In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A(k) in L2(- 1, 1), generated from the classical second-order Legendre differential equation lL,k[y](t) = -((1 - t2)y')' + ky = λy (t∈ (-1,1)), that has the Legendre polynomials {Pm(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 Wn(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 An(k) in Wn(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 lL,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.