Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

Abstract

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family { ] ( )} ∞ ]=0 orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations A ] ( ) = ] ( ) ] ( ), where A is a linear differential operator and each ] ( ) is a polynomial of degree at most 0 ∈ N; 0 does not depend on ]. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

Solvable dynamical systems and isospectral matrices defined in terms of the zeros of orthogonal or otherwise special polynomials

“…Finding similarity matrices for each isospectral matrix L in Propositions 2, 4, 5, 6, 7, 8 would provide a tool for construction of additional algebraic identities satisfied by the zeros of the appropriate polynomials. A method that compares the spectral and the pseudospectral matrix representations of linear differential operators has been employed in [4,7,16] to construct new and re-obtain known algebraic identities satisfied by classical, Krall and Sonin-Markov orthogonal polynomials. It would be interesting to adapt the last method to the cases of the generalized hypergeometric, generalized basic hypergeometric, Wilson and Racah as well as Askey-Wilson and q-Racah polynomials and to compare the results with those reviewed in this chapter as well as to utilize the new developments to study properties of the solvable nonlinear systems (45), (68), (87), (100), (117), (127).…”

“…Recall that the operator ∆ γ does not raise degrees of polynomials when acting on them, see the remark following display (16). Also, the operator ∆ 1 annihilates functions independent of z, while the operator ∆ q −N annihilates z N .…”

“…Since the groundbreaking work of Stieltjes, the literature on algebraic relations satisfied by the zeros of orthogonal or otherwise special polynomials is abundant, see for example [3,1,2,19,20,38,4,8,9,10,11,7,16].…”