On the zeros of 𝑑-orthogonal Laguerre polynomials and their 𝑞-analogues

Ben Romdhane, Neila

doi:10.1090/proc/13164

On the zeros of $d$-orthogonal Laguerre polynomials and their $q$-analogues
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Abstract:

In this paper, we give some properties of the zeros of special families of Brenke type polynomials. In particular, we consider an extension of Laguerre polynomials known as $d$-orthogonal Laguerre polynomials. For these polynomials, we prove that all the zeros are simple, positive and interlaced. A $q$-analogue is considered as well.

References

A. I. Aptekarev, Multiple orthogonal polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 1998, pp. 423–447. MR 1662713, DOI 10.1016/S0377-0427(98)00175-7
A. Aptekarev, V. Kaliaguine, and J. Van Iseghem, The genetic sums’ representation for the moments of a system of Stieltjes functions and its application, Constr. Approx. 16 (2000), no. 4, 487–524. MR 1771693, DOI 10.1007/s003650010004
A. I. Aptekarev, V. A. Kalyagin, and E. B. Saff, Higher-order three-term recurrences and asymptotics of multiple orthogonal polynomials, Constr. Approx. 30 (2009), no. 2, 175–223. MR 2519660, DOI 10.1007/s00365-008-9032-0
Youssef Ben Cheikh and Neila Ben Romdhane, $d$-symmetric $d$-orthogonal polynomials of Brenke type, J. Math. Anal. Appl. 416 (2014), no. 2, 735–747. MR 3188736, DOI 10.1016/j.jmaa.2014.02.046
Youssèf Ben Cheikh and Khalfa Douak, On two-orthogonal polynomials related to the Bateman’s $J_n^{u,v}$-function, Methods Appl. Anal. 7 (2000), no. 4, 641–662. MR 1868550, DOI 10.4310/MAA.2000.v7.n4.a3
Y. Ben Cheikh, I. Lamiri, and A. Ouni, $d$-orthogonality of little $q$-Laguerre type polynomials, J. Comput. Appl. Math. 236 (2011), no. 1, 74–84. MR 2823415, DOI 10.1016/j.cam.2011.03.006
N. Ben Romdhane, On the zeros of $d$-symmetric $d$-orthogonal polynomials, J. Math. Anal. Appl. 344 (2008), no. 2, 888–897. MR 2426317, DOI 10.1016/j.jmaa.2008.02.038
Fred Brafman, Generating functions of Jacobi and related polynomials, Proc. Amer. Math. Soc. 2 (1951), 942–949. MR 45875, DOI 10.1090/S0002-9939-1951-0045875-2
W. C. Brenke, On generating functions of polynomial systems, Amer. Math. Monthly 52 (1945), 297–301. MR 12720, DOI 10.2307/2305289
Khalfa Douak and Pascal Maroni, Les polynômes orthogonaux “classiques” de dimension deux, Analysis 12 (1992), no. 1-2, 71–107 (French, with English summary). MR 1159372, DOI 10.1524/anly.1992.12.12.71
Dharma P. Gupta and Martin E. Muldoon, Inequalities for the smallest zeros of Laguerre polynomials and their $q$-analogues, JIPAM. J. Inequal. Pure Appl. Math. 8 (2007), no. 1, Article 24, 7. MR 2295718
Maciej Haneczok and Walter Van Assche, Interlacing properties of zeros of multiple orthogonal polynomials, J. Math. Anal. Appl. 389 (2012), no. 1, 429–438. MR 2876509, DOI 10.1016/j.jmaa.2011.11.077
Arno B. J. Kuijlaars and Lun Zhang, Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits, Comm. Math. Phys. 332 (2014), no. 2, 759–781. MR 3257662, DOI 10.1007/s00220-014-2064-3
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
Pascal Maroni, L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux, Ann. Fac. Sci. Toulouse Math. (5) 10 (1989), no. 1, 105–139 (French, with English and French summaries). MR 1425747
Daniel S. Moak, The $q$-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), no. 1, 20–47. MR 618759, DOI 10.1016/0022-247X(81)90048-2
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
Javier Segura, The zeros of special functions from a fixed point method, SIAM J. Numer. Anal. 40 (2002), no. 1, 114–133. MR 1921912, DOI 10.1137/S0036142901387385
W. Van Assche, Ratio asymptotics for multiple orthogonal polynomials, Contemp. Math. (arXiv: 1408.1829), To appear.
Walter Van Assche, Galina Filipuk, and Lun Zhang, Multiple orthogonal polynomials associated with an exponential cubic weight, J. Approx. Theory 190 (2015), 1–25. MR 3304586, DOI 10.1016/j.jat.2014.06.006
J. Van Iseghem, Vector orthogonal relations. Vector QD-algorithm, J. Comput. Appl. Math. 19 (1987), no. 1, 141–150. MR 901223, DOI 10.1016/S0377-0427(87)80019-5