Abstract
We introduce a large class of measures with orthogonal polynomials satisfying higher-order difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights of Charlier, Meixner, Krawtchouk, and Hahn by certain variants of the annihilator polynomial of a finite set of numbers.
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Álvarez-Nodarse, R., Durán, A.J.: Using \(\mathcal{D}\)-operators to construct orthogonal polynomials satisfying higher order q-difference equations (in preparation)
Bavinck, H., van Haeringen, H.: Difference equations for generalizations of Meixner polynomials. J. Math. Anal. Appl. 184, 453–463 (1994)
Bavinck, H., Koekoek, R.: On a difference equation for generalizations of Charlier polynomials. J. Approx. Theory 81, 195–206 (1995)
Bochner, S.: Über Sturm–Liouvillesche Polynomsysteme. Math. Z. 29, 730–736 (1929)
Bueno, M.I., Marcellán, F.: Darboux transformation and perturbation of linear functionals. Linear Algebra Appl. 384, 215–242 (2004)
Christoffel, E.B.: Über die Gaussische quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math. 55, 61–82 (1858)
Durán, A.J.: The algebra of difference operators associated to a family of orthogonal polynomials. J. Approx. Theory 164, 586–610 (2012)
Durán, A.J.: How to use \(\mathcal{D}\)-operators to find orthogonal polynomials satisfying higher order differential equations (in preparation)
Durán, A.J.: Using \(\mathcal {D}\)-operators to construct orthogonal polynomials satisfying higher order difference equations (in preparation)
Geronimus, Ya.L.: On the polynomials orthogonal with respect to a given number sequence and a theorem by W. Hahn. Izv. Akad. Nauk SSSR, Ser. Mat. 4, 215–228 (1940)
Gómez-Ullate, D., Kamran, N., Milson, R.: An extended class of orthogonal polynomials defined by a Sturm–Liouville problem. J. Math. Anal. Appl. 359, 352–367 (2009)
Grünbaum, F.A., Haine, L.: Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation. In: Levi, D., Vinet, L., Winternitz, P. (eds.) Symmetries an Integrability of Differential Equations. CRM Proc. Lecture Notes, vol. 9, pp. 143–154. Am. Math. Soc., Providence (1996)
Grünbaum, F.A., Haine, L.: The q-version of a theorem of Bochner. J. Comput. Appl. Math. 68, 103–114 (1996)
Grünbaum, F.A., Haine, L., Horozov, E.: Some functions that generalize the Krall–Laguerre polynomials. J. Comput. Appl. Math. 106, 271–297 (1999)
Grünbaum, F.A., Yakimov, M.: Discrete bispectral Darboux transformations from Jacobi operators. Pac. J. Math. 204, 395–431 (2002)
Koekoek, R.: Differential equations for symmetric generalized ultraspherical polynomials. Trans. Am. Math. Soc. 345, 47–72 (1994)
Koekoek, J., Koekoek, R.: On a differential equation for Koornwinder’s generalized Laguerre polynomials. Proc. Am. Math. Soc. 112, 1045–1054 (1991)
Koekoek, J., Koekoek, R.: Differential equations for generalized Jacobi polynomials. J. Comput. Appl. Math. 126, 1–31 (2000)
Koornwinder, T.H.: Orthogonal polynomials with weight function (1−x)α(1+x)β+Mδ(x+1)+Nδ(x−1). Can. Math. Bull. 27, 205–214 (1984)
Krall, H.L.: Certain differential equations for Tchebycheff polynomials. Duke Math. J. 4, 705–718 (1938)
Krall, H.L.: On orthogonal polynomials satisfying a certain fourth order differential equation. The Pennsylvania State College Studies, 6 (1940)
Krall, A.M., Littlejohn, L.L.: On the classification of differential equations having orthogonal polynomial solutions. II. Ann. Mat. Pura Appl. 149, 77–102 (1987)
Kwon, K.H., Lee, D.W.: Characterizations of Bochner–Krall orthogonal polynomials of Jacobi type. Constr. Approx. 19, 599–619 (2003)
Kwon, K.H., Littlejohn, L.L., Yoon, G.J.: Orthogonal polynomial solutions of spectral type differential equations: Magnus’ conjecture. J. Approx. Theory 112, 189–215 (2001)
Kwon, K.H., Littlejohn, L.L., Yoon, G.J.: Construction of differential operators having Bochner–Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324, 285–303 (2006)
Lancaster, O.E.: Orthogonal polynomials defined by difference equations. Am. J. Math. 63, 185–207 (1941)
Littlejohn, L.: The Krall polynomials: a new class of orthogonal polynomials. Quaest. Math. 5, 255–265 (1982)
Littlejohn, L.: On the classification of differential equations having orthogonal polynomial solutions. Ann. Mat. Pura Appl. 93, 35–53 (1984)
Littlejohn, L.: An application of a new theorem on orthogonal polynomials and differential equations. Quaest. Math. 10, 49–61 (1986)
Maroni, P.: Tchebychev forms and their perturbed forms as second degree forms. Ann. Numer. Math. 2, 123–143 (1995)
Sasaki, R., Tsujimoto, S., Zhedanov, A.: Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations. J. Phys. A, Math. Gen. 43, 315204 (2010)
Szegö, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1959)
Vinet, L., Zhedanov, A.: Generalized little q-Jacobi polynomials as eigensolutions of higher-order q-difference operators. Proc. Am. Math. Soc. 129, 1317–1327 (2001)
Yermolayeva, O., Zhedanov, A.: Spectral transformations and generalized Pollaczek polynomials. Methods Appl. Anal. 6, 261–280 (1999)
Yoon, G.J.: Darboux transforms and orthogonal polynomials. Bull. Korean Math. Soc. 39, 359–376 (2002)
Zhedanov, A.: A method of constructing Krall’s polynomials. J. Comput. Appl. Math. 107, 1–20 (1999)
Acknowledgements
The author would like to thank T. Koornwinder and two unknown referees for providing important references and R. Alvarez-Nodarse for some fruitful discussions while preparing this paper.
Partially supported by D.G.E.S, ref. MTM2009-12740-C03-02, FQM-262, FQM-4643 (Junta de Andalucía).
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Communicated by Tom H. Koornwinder.
Dedicated to the memory of Alejandro Fernández-Margarit.
Appendix
Appendix
The purpose of this paper is not to study the structural properties of the orthogonal polynomials introduced above. Nevertheless, we would like to finish with a couple of comments about that issue.
Firstly, we point out here that they can be defined by means of modified Rodrigues’ formulas. Since their orthogonalizing measures are defined by applying a Christoffel transform to the discrete classical measures, we can also get expressions for these polynomials in terms of the discrete classical polynomials. Grünbaum and Haine (et al.) proved that polynomials satisfying fourth- or higher-order differential equations can be generated by applying a Darboux transform to certain choices of the classical polynomials (see [12, 14, 15]). That is also the case of our polynomials: one can obtain them by one (or several) Darboux transforms applied to certain instances of the discrete classical polynomials. Here is an example to illustrate this last statement.
The formula
defines a sequence of orthogonal polynomials with respect to the measure ρ=∑ x∈ℕ ρ(x)δ x defined by (1.11).
The above Rodrigues’ formula allows us to expand the orthogonal polynomials p n , n≥0, in terms of the Meixner polynomials. Indeed, write \(\omega_{a,c} (x)=\frac{a^{x}\varGamma(x+c)}{x!}\) so that ω a,c =∑ x∈ℕ ω a,c (x)δ x is the Meixner measure (1.6), and consider the following Rodrigues’ formula for Meixner polynomials:
It is then easy to see that the polynomials p n , n≥1, are equal to
Since the leading coefficient of m n,a,c is (a−1)n, we get that the polynomials
are monic.
Since ρ=(x+c+1)ω a,c , we have that ρ is a Christoffel transform of the Meixner measure ω a,c . Using (2.5), we then have
where
Assume that the Jacobi matrix J associated with the sequence of monic orthogonal polynomials with respect to a measure μ can be decomposed as
where
The three-diagonal matrix \(\tilde{J}=BA+\lambda I\) is then called a Darboux transform of J. If \(\tilde{J}_{i+1,i}\not=0\) for i=1,2,…, then \(\tilde{J}\) is also the Jacobi matrix associated with a certain measure \(\tilde{\mu}\). It was proved by G.J. Yoon that the measures μ and \(\tilde{\mu}\) are related by the formula \((x-\lambda)\tilde{\mu}=\mu\) (see [35], Theorem 2.4). The measure \(\tilde{\mu}\) is sometimes called a Geronimus transform of the measure μ (Geronimus transform is reciprocal to the Christoffel transform; see [10] or, for a modern systematic treatment, [5, 30]).
Since the measure ρ (1.11) can be written in the form ρ=ω a,c+2/(x+c), we have that the orthogonal polynomials (p n ) n with respect to ρ can be obtained by applying a Darboux transform to the Meixner polynomials m n,c+2. The explicit expressions for the matrices A and B in this case are
From this we can get the three-term recurrence coefficients for the sequence (p n ) n :
where
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Durán, A.J. Orthogonal Polynomials Satisfying Higher-Order Difference Equations. Constr Approx 36, 459–486 (2012). https://doi.org/10.1007/s00365-012-9162-2
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DOI: https://doi.org/10.1007/s00365-012-9162-2
Keywords
- Orthogonal polynomials
- Difference equations
- Difference operators
- Discrete classical polynomials
- Charlier polynomials
- Meixner polynomials
- Krawtchouk polynomials
- Hahn polynomials