zbMATH — the first resource for mathematics

Laguerre polynomials generalized to a certain discrete Sobolev inner product space. (English) Zbl 0771.42015
Summary: We are concerned with the set of polynomials \(\{S_ n^{M,N}\}\) which are orthogonal with respect to the discrete Sobolev inner product \[ \langle f,g\rangle=\int^ \infty_ 0w(x)f(x)g(x)dx+Mf(0)g(0)+Nf'(0)g'(0), \] where \(w\) is a weight function, \(M\geq 0\), \(N\geq 0\). We show that these polynomials can be described as a linear combination of standard polynomials which are orthogonal with respect to the weight functions \(w(x)\), \(x^ 2w(x)\), and \(x^ 4w(x)\). The location of the zeros of \(S_ n^{M,N}\) is given in relation to the position of the zeros of the standard polynomials.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI