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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.

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.)
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