On differential equations for Sobolev-type Laguerre polynomials

Authors:
J. Koekoek, R. Koekoek and H. Bavinck

Journal:
Trans. Amer. Math. Soc. **350** (1998), 347-393

MSC (1991):
Primary 33C45; Secondary 34A35

DOI:
https://doi.org/10.1090/S0002-9947-98-01993-X

MathSciNet review:
1433121

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Abstract | References | Similar Articles | Additional Information

Abstract: The Sobolev-type Laguerre polynomials $\{L_n^{\alpha ,M,N}(x)\}_{n=0}^{\infty }$ are orthogonal with respect to the inner product \[ \langle f,g\rangle \;=\frac {1}{\Gamma (\alpha +1)}\int _0^{\infty }x^{\alpha }e^{-x}f(x)g(x)dx+Mf(0)g(0)+ Nf’(0)g’(0),\] where $\alpha >-1$, $M\ge 0$ and $N\ge 0$. In 1990 the first and second author showed that in the case $M>0$ and $N=0$ the polynomials are eigenfunctions of a unique differential operator of the form \[ M\sum _{i=1}^{\infty }a_i(x)D^i+xD^2+(\alpha +1-x)D,\] where $\left \{a_i(x)\right \}_{i=1}^{\infty }$ are independent of $n$. This differential operator is of order $2\alpha +4$ if $\alpha$ is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form \begin{align} &M\sum _{i=0}^{\infty }a_i(x)y^{(i)}(x)+ N\sum _{i=0}^{\infty }b_i(x)y^{(i)}(x)\\ &\hspace {1cm}{}+MN\sum _{i=0}^{\infty }c_i(x)y^{(i)}(x)+ xy''(x)+(\alpha +1-x)y’(x)+ny(x)=0, \end{align} where the coefficients $\left \{a_i(x)\right \}_{i=1}^{\infty }$, $\left \{b_i(x)\right \}_{i=1}^{\infty }$ and $\left \{c_i(x)\right \}_{i=1}^{\infty }$ are independent of $n$ and the coefficients $a_0(x)$, $b_0(x)$ and $c_0(x)$ are independent of $x$, satisfied by the Sobolev-type Laguerre polynomials $\{L_n^{\alpha ,M,N}(x)\}_{n=0}^{\infty }$. Further, we show that in the case $M=0$ and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order $2\alpha +8$ if $\alpha$ is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case $M>0$ and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order $4\alpha +10$ if $\alpha$ is a nonnegative integer and of infinite order otherwise.

- H. Bavinck,
*A direct approach to Koekoek’s differential equation for generalized Laguerre polynomials*, Acta Math. Hungar.**66**(1995), no. 3, 247–253. MR**1314004**, DOI https://doi.org/10.1007/BF01874288 - H. Bavinck :
*A difference operator of infinite order with Sobolev-type Charlier polynomials as eigenfunctions.*Indagationes Mathematicae, (N.S.)**7**(3), 1996, 281–291. - Herman Bavinck and Henk van Haeringen,
*Difference equations for generalized Meixner polynomials*, J. Math. Anal. Appl.**184**(1994), no. 3, 453–463. MR**1281521**, DOI https://doi.org/10.1006/jmaa.1994.1214 - Herman Bavinck and Roelof Koekoek,
*On a difference equation for generalizations of Charlier polynomials*, J. Approx. Theory**81**(1995), no. 2, 195–206. MR**1327166**, DOI https://doi.org/10.1006/jath.1995.1044 - W. N. Everitt, L. L. Littlejohn, and R. Wellman,
*The symmetric form of the Koekoeks’ Laguerre type differential equation*, Proceedings of the Fourth International Symposium on Orthogonal Polynomials and their Applications (Evian-Les-Bains, 1992), 1995, pp. 115–121. MR**1340929**, DOI https://doi.org/10.1016/0377-0427%2893%29E0238-H - I.H. Jung, K.H. Kwon, D.W. Lee & L.L. Littlejohn :
*Sobolev orthogonal polynomials and spectral differential equations.*Transactions of the American Mathematical Society**347**, 1995, 3629-3643. - I. H. Jung, K. H. Kwon, D. W. Lee, and L. L. Littlejohn,
*Differential equations and Sobolev orthogonality*, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 173–180. MR**1379129**, DOI https://doi.org/10.1016/0377-0427%2895%2900111-5 - J. Koekoek and R. Koekoek,
*On a differential equation for Koornwinder’s generalized Laguerre polynomials*, Proc. Amer. Math. Soc.**112**(1991), no. 4, 1045–1054. MR**1047003**, DOI https://doi.org/10.1090/S0002-9939-1991-1047003-9 - R. Koekoek :
*Generalizations of the classical Laguerre polynomials and some q-analogues.*Delft University of Technology, Thesis, 1990. - R. Koekoek :
*The search for differential equations for orthogonal polynomials by using computers.*Delft University of Technology, report no.**91-55**, 1991. - Roelof Koekoek,
*The search for differential equations for certain sets of orthogonal polynomials*, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 111–119. MR**1256017**, DOI https://doi.org/10.1016/0377-0427%2893%2990141-W - R. Koekoek and H. G. Meijer,
*A generalization of Laguerre polynomials*, SIAM J. Math. Anal.**24**(1993), no. 3, 768–782. MR**1215437**, DOI https://doi.org/10.1137/0524047 - Tom H. Koornwinder,
*Orthogonal polynomials with weight function $(1-x)^{\alpha }(1+x)^{\beta }+M\delta (x+1)+N\delta (x-1)$*, Canad. Math. Bull.**27**(1984), no. 2, 205–214. MR**740416**, DOI https://doi.org/10.4153/CMB-1984-030-7 - Allan M. Krall,
*Orthogonal polynomials satisfying fourth order differential equations*, Proc. Roy. Soc. Edinburgh Sect. A**87**(1980/81), no. 3-4, 271–288. MR**606336**, DOI https://doi.org/10.1017/S0308210500015213 - H.L. Krall :
*Certain differential equations for Tchebycheff polynomials.*Duke Mathematical Journal**4**, 1938, 705-718. - H.L. Krall :
*On orthogonal polynomials satisfying a certain fourth order differential equation.*The Pennsylvania State College Studies, No. 6, 1940. - H. L. Krall and I. M. Sheffer,
*Differential equations of infinite order for orthogonal polynomials*, Ann. Mat. Pura Appl. (4)**74**(1966), 135–172. MR**206441**, DOI https://doi.org/10.1007/BF02416454 - Yudell L. Luke,
*The special functions and their approximations. Vol. II*, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR**0249668** - C.S. Meijer :
*Expansion theorems for the $G$-function I.*Indagationes Mathematicae**14**, 1952, 369-379.

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Additional Information

**J. Koekoek**

Affiliation:
Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600 GA Delft, The Netherlands

**R. Koekoek**

Affiliation:
Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600 GA Delft, The Netherlands

Email:
r.koekoek@twi.tudelft.nl

**H. Bavinck**

Affiliation:
Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600 GA Delft, The Netherlands

Email:
h.bavinck@twi.tudelft.nl

Keywords:
Differential equations,
Sobolev-type Laguerre polynomials

Received by editor(s):
August 28, 1995

Received by editor(s) in revised form:
June 24, 1996

Article copyright:
© Copyright 1998
American Mathematical Society