A DIFFERENCE EQUATION FOR MULTIPLE KRAVCHUK POLYNOMIALS

Title & Authors
A DIFFERENCE EQUATION FOR MULTIPLE KRAVCHUK POLYNOMIALS
Lee, Dong-Won;

Abstract
Let $\small{{K^{(\vec{p};N)}_{\vec{n}}(x)}}$ be a multiple Kravchuk polynomial with respect to r discrete Kravchuk weights. We first find a lowering operator for multiple Kravchuk polynomials $\small{{K^{(\vec{p};N)}_{\vec{n}}(x)}}$ in which the orthogonalizing weights are not involved. Combining the lowering operator and the raising operator by Rodrigues# formula, we find a (r+1)-th order difference equation which has the multiple Kravchuk polynomials $\small{{K^{(\vec{p};N)}_{\vec{n}}(x)}}$ as solutions. Lastly we give an explicit difference equation for $\small{{K^{(\vec{p};N)}_{\vec{n}}(x)}}$ for the case of r
Keywords
multiple orthogonal polynomials;Kravchuk polynomials;difference equation;rodrigues` formula;
Language
English
Cited by
1.
Difference equations for discrete classical multiple orthogonal polynomials, Journal of Approximation Theory, 2008, 150, 2, 132
References
1.
A. Angelesco, Sur l'approximation simultanee de plusieurs integrales definies, C. R. Paris, 167 (1918), 629-631

2.
A. I. Aptekarev, Multiple orthogonal polynomials, J. Comp. Appl. Math. 99 (1998), no. 1-2, 423-447

3.
A. I. Aptekarev, A. Branquinho, and W. Van Assche, Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3887-3914

4.
A. I. Aptekarev and H. Stahl, Asymptotics of Hermite-Pade polynomials, in A. Gonchar and E. B. Saff (Eds.), Progress in Approximation Theory, vol. 19, Springer Ser. Comp. Math. Springer, (1992), 127-167

5.
J. Arvesu, J. Coussement, and W. Van Assche, Some discrete multiple orthogonal poly- nomials, J. Comp. Appl. Math. 153 (2003), no. 1-2, 19-45

6.
B. Beckermann, J. Coussement, and W. Van Assche, Multiple Wilson and Jacobi-Pineiro polynomials, J. Approx. Theory 132 (2005), no. 2, 155-181

7.
C. Brezinski and J. Van Iseghem, Vector orthogonal polynomials of dimension -d, Ap- proximation and computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., 119, Birkhauser Boston, Boston, (1994), 29-39

8.
M. G. de Bruin, Simultaneous Pade approximants and orthogonality, Lecture Notes in Math. 1171, Springer, (1985), 74-83

9.
J. Coussement and W. Van Assche, Gaussian quadrature for multiple orthogonal poly- nomials, J. Comp. Appl. Math. 178 (2005), no. 1-2, 131-145

10.
V. A. Kalyagin, Hermite-Pade approximants and spectral analysis of nonsymmetric operators, Math. Sb. 185 (1994), no. 6, 79-100

11.
V. A. Kalyagin, Hermite-Pade approximants and spectral analysis of nonsymmetric operators, English transl. in. Russian Acad. Sci. Sb. Math. 82 (1995), no. 1, 199-216

12.
M. Krawtchouk, Sur Une Generalisation des Polynomes d'Hermite, C. R. Acad. Sci. 189 (1929), 620-622

13.
D. W. Lee, Some recurrence relations of multiple orthogonal polynomials, J. Korean Math. Soc. 42 (2005), no. 4, 673-693

14.
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag, Berlin, 1991

15.
E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality, Trans- lations of Mathematical Monographs 92, Amer. Math. Soc., 1991

16.
K. Postelmans and W. Van Assche, Multiple little q-Jacobi polynomials, J. Comput. Appl. Math. 178 (2005), no. 1-2, 361-375

17.
G. Szego, Orthogonal Polynomials, 4th ed., Amer. Math. Soc., Colloq. Publ. 23, Providence, RI, 1975

18.
W. Van Assche and E. Coussement, Some classical multiple orthogonal polynomials, J. Comp. Appl. Math. 127 (2001), no. 1-2, 317-347

19.
W. Van Assche, Difference equations for multiple Charlier and Meixner polynomials, in New Progress in Difference Equations (S. Elaydi et al. eds.), Taylor and Francis, London, (2004), 547-557

20.
J. Van Iseghem, Recurrence relations in the table of vector orthogonal polynomials, Nonlinear Numerical Methods and Rational Approximation II, Math. Appl., Kluwer Academic Publishers, Dordrecht, 296 (1994), 61-69