zbMATH — the first resource for mathematics

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 (*).

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI
[1] Bavinck, H; Meijer, H.G, Orthogonal polynomials with respect to a symmetric inner product, Appl. anal., 33, 103-117, (1989) · Zbl 0648.33007
[2] Chihara, T.S, An introduction to orthogonal polynomials, () · Zbl 0389.33008
[3] Koekoek, J; Koekoek, R, A simple proof of a differential equation for generalizations of Laguerre polynomials, Delft university of technology, faculty of mathematics and informatics, report no. 89-15, (1989) · Zbl 0737.33003
[4] \scR. Koekoek and H. G. Meijer, “A generalization of Laguerre Polynomials,” Delft University of Technology, Faculty of Mathematics and Informatics, report no. 88-28, to appear in SIAM J. Math. Anal. · Zbl 0780.33007
[5] Koekoek, R, Koornwinder’s Laguerre polynomials, Delft progr. rep., 393-404, (1988) · Zbl 0657.33008
[6] Koornwinder, T.H, Orthogonal polynomials with weight function (1 − x)α(1 + x)β + M · δ(x + 1) + N · δ(x − 1), Canad. math. bull., 27, No. 2, 205-214, (1984) · Zbl 0507.33005
[7] Krall, A.M, Orthogonal polynomials satisfying fourth order differential equations, (), 271-288 · Zbl 0453.33006
[8] Krall, H.L, Certain differential equations for Tchebycheff polynomials, Duke math. J., 4, 705-718, (1938) · Zbl 0020.02002
[9] Krall, H.L, On orthogonal polynomials satisfying a certain fourth order differential equation, The pennsylvania state college studies, no. 6, (1940) · Zbl 0060.19210
[10] Szegö, G, Orthogonal polynomials, () · JFM 65.0278.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.