### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 026, 14 pages      arXiv:1411.2000      https://doi.org/10.3842/SIGMA.2015.026
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### On the $q$-Charlier Multiple Orthogonal Polynomials

Jorge Arvesú and Andys M. Ramírez-Aberasturis

Received November 10, 2014, in final form March 23, 2015; Published online March 28, 2015

Abstract
We introduce a new family of special functions, namely $q$-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to $q$-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a $q$-analogue of the second of Appell's hypergeometric functions is given. A high-order linear $q$-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.

Key words: multiple orthogonal polynomials; Hermite-Padé approximation; difference equations; classical orthogonal polynomials of a discrete variable; Charlier polynomials; $q$-polynomials.

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References

1. Álvarez-Nodarse R., Arvesú J., On the $q$-polynomials in the exponential lattice $x(s)=c_1q^s+c_3$, Integral Transform. Spec. Funct. 8 (1999), 299-324.
2. Aptekarev A., Arvesú J., Asymptotics for multiple Meixner polynomials, J. Math. Anal. Appl. 411 (2014), 485-505, arXiv:1207.0463.
3. Arvesú J., On some properties of $q$-Hahn multiple orthogonal polynomials, J. Comput. Appl. Math. 233 (2010), 1462-1469.
4. Arvesú J., Coussement J., Van Assche W., Some discrete multiple orthogonal polynomials, J. Comput. Appl. Math. 153 (2003), 19-45.
5. Arvesú J., Esposito C., A high-order $q$-difference equation for $q$-Hahn multiple orthogonal polynomials, J. Difference Equ. Appl. 18 (2012), 833-847, arXiv:0910.4041.
6. Borodin A., Ferrari P.L., Sasamoto T., Two speed TASEP, J. Stat. Phys. 137 (2009), 936-977, arXiv:0904.4655.
7. Daems E., Kuijlaars A.B.J., A Christoffel-Darboux formula for multiple orthogonal polynomials, J. Approx. Theory 130 (2004), 190-202, math.CA/0402031.
8. Ernst T., On the $q$-analogues of Srivastava's triple hypergeometric functions, Axioms 2 (2013), 85-99.
9. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
10. Haneczok M., Van Assche W., Interlacing properties of zeros of multiple orthogonal polynomials, J. Math. Anal. Appl. 389 (2012), 429-438, arXiv:1108.3917.
11. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
12. Lee D.W., Difference equations for discrete classical multiple orthogonal polynomials, J. Approx. Theory 150 (2008), 132-152.
13. Miki H., Vinet L., Zhedanov A., Non-Hermitian oscillator Hamiltonians and multiple Charlier polynomials, Phys. Lett. A 376 (2011), 65-69.
14. Ndayiragije F., Van Assche W., Asymptotics for the ratio and the zeros of multiple Charlier polynomials, J. Approx. Theory 164 (2012), 823-840, arXiv:1108.3918.
15. Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.
16. Nikishin E.M., Sorokin V.N., Rational approximations and orthogonality, Translations of Mathematical Monographs, Vol. 92, Amer. Math. Soc., Providence, RI, 1991.
17. Postelmans K., Van Assche W., Multiple little $q$-Jacobi polynomials, J. Comput. Appl. Math. 178 (2005), 361-375, math.CA/0403532.
18. Prévost M., Rivoal T., Remainder Padé approximants for the exponential function, Constr. Approx. 25 (2007), 109-123.
19. Van Assche W., Difference equations for multiple Charlier and Meixner polynomials, in Proceedings of the Sixth International Conference on Difference Equations, CRC, Boca Raton, FL, 2004, 549-557.
20. Van Assche W., Nearest neighbor recurrence relations for multiple orthogonal polynomials, J. Approx. Theory 163 (2011), 1427-1448, arXiv:1104.3778.