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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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