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Generalizations of Laguerre polynomials. (English) Zbl 0737.33004
The main aim of the author is to show that the constants \(A_ 0,A_ 1,\dots,A_{N+1}\) can appropriately be chosen such that the polynomials \[ L_ n^{\alpha,M_ 0,M_ 1,\dots,M_ n}(x)=\sum_{k=0}^{N+1} A_ k D^ k L_ n^{(\alpha)}(x); \qquad \alpha>-1; \quad n=0,1,\dots (*) \] constitute an orthogonal set with respect to the following inner product: \[ \langle f,g\rangle ={1 \over \Gamma(\alpha+1)}\int_ 0^ \infty x^ \alpha e^{-x}f(x)g(x)dx+\sum_{\nu=0}^ N M_ \nu f_{(0)}^{(\nu)}g_{(0)}^{(\nu)}. (**) \] In (*) \(L_ n^{(\alpha)}(x)\) stands for the classical Laguerre polynomial while in (**) \(M_ \nu\) are certain given nonnegative constants. The most appealing feature of the result is that the constants \(A_ k\) are independent of the index \(n\). He derives also a second order differential equation with polynomial coefficients as well as a \((2N+3)\)-terms recurrence relation satisfied by the new polynomials given by (*).

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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