Laguerre-Sobolev orthogonal polynomials
The authors consider the polynomials \(Q_n^{(\alpha)} (x)\) \((n= 0, 1, \dots)\) which are orthogonal with respect to the inner product \[ (f, g)_S= \int_0^{+\infty} f(x) g(x) x^\alpha e^{-x} dx+ \lambda \int_0^{+\infty} f' (x) g' (x) x^\alpha e^{-x} dx, \tag \(*\) \] and satisfy the relations \[ (Q_n^{(\alpha)}, x^m)_S= 0, \qquad m=0, 1, \dots, n-1. \tag \(**\) \] Here one supposes \(\alpha> -1\), \(\lambda\geq 0\) and \(Q_0^{\alpha} (x)=1\). The case \(\lambda =0\) gives the classical Laguerre polynomials \(L_n ^{(\alpha)} (x)\). Their aim consists of deriving several relations which are satisfied by these polynomials themselves or connect these polynomials to the classical Laguerre polynomials (four-term recurrence relation, difference-differential relation, Christoffel-Darboux-type relation, Rodrigues-type relation, etc). The zeros of the polynomials \(Q_n^{(\alpha)} (x)\) and their localizations with respect to the zeros of the Laguerre polynomials are also studied. The paper involves 22 Lemmas, theorems, propositions and corollaries, and indicates some open problems.
scientific article; zbMATH DE number 910315