Laguerre polynomials generalized to a certain discrete Sobolev inner product space. (English) Zbl 0771.42015

Meijer, H. G.

J. Approximation Theory 73, No. 1, 1-16 (1993).

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.

MSC:

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.)

Keywords:

orthogonal polynomials; Laguerre polynomials; discrete Sobolev inner product; weight function; location of the zeros