On the Spectral Analysis of the Laguerre Polynomials Ln−k(x) for Positive Integers kLn˜−k(x) for Positive Integers k

This paper discusses various spectral representations of the Laguerre polynomial sequences https://www.w3.org/1998/Math/MathML">Lnα(x)n=−α∞https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/ieq0493.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML">Lnα(x)n∞≡0https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/ieq0494.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> when the parameter α is a negative integer. Besides a discussion of the right-definite and left-definite spectral theory of the Laguerre sequence https://www.w3.org/1998/Math/MathML">Lnα(x)n=−α∞https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/ieq0495.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for all −α ∈ ℕ, we discuss spectral properties of the entire Laguerre sequence https://www.w3.org/1998/Math/MathML">Lnα(x)n=−α∞https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/ieq0496.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> when α = −1 and α = −2, in a weighted Sobolev space (W_∣α∣,(·, ·)w∣_α∣), where these Laguerre polynomials form a complete orthonormal sequence. The inner product (·, ·)∣_α∣, for any −α ∈ ℕ, was recently discovered by K won and Littlejohn. In all three settings, we shall show that these Laguerre polynomials form a complete set of eigenfunctions of a self-adjoint differential operator with discrete spectrum generated from the Laguerre differential expression https://www.w3.org/1998/Math/MathML"> l α [ y ] ( x ) = − x − α e x x α + 1 e − x y ′ ( x ) ′ + y ( x )   ( x ∈ ( 0 , ∞ ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/eqn0520.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>