Geodesic
Solution of an Open Problem about Two Families of Orthogonal Polynomials
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An open problem about two new families of orthogonal polynomials was posed by Alhaidari. Here we will identify one of them as Wilson polynomials. The other family seems to be new but we show that they are discrete orthogonal polynomials on a bounded countable set with one accumulation point at $0$ and we give some asymptotics as the degree tends to infinity.
Keywords: orthogonal polynomials; special functions; open problems. 
@article{SIGMA_2019_15_a4,
     author = {Walter Van Assche},
     title = {Solution of an {Open} {Problem} about {Two} {Families} of {Orthogonal} {Polynomials}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a4/}
}
TY  - JOUR
AU  - Walter Van Assche
TI  - Solution of an Open Problem about Two Families of Orthogonal Polynomials
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2019
VL  - 15
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a4/
LA  - en
ID  - SIGMA_2019_15_a4
ER  - 
%0 Journal Article
%A Walter Van Assche
%T Solution of an Open Problem about Two Families of Orthogonal Polynomials
%J Symmetry, integrability and geometry: methods and applications
%D 2019
%V 15
%U http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a4/
%G en
%F SIGMA_2019_15_a4
Walter Van Assche. Solution of an Open Problem about Two Families of Orthogonal Polynomials. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a4/

[1] Alhaidari A. D., “An extended class of $L^2$-series solutions of the wave equation”, Ann. Physics, 317 (2005), 152–174, arXiv: quant-ph/0409002 | DOI | MR | Zbl

[2] Alhaidari A. D., “Solution of the nonrelativistic wave equation using the tridiagonal representation approach”, J. Math. Phys., 58 (2017), 072104, 37 pp., arXiv: 1703.01268 | DOI | MR | Zbl

[3] Alhaidari A. D., “Orthogonal polynomials derived from the tridiagonal representation approach”, J. Math. Phys., 59 (2018), 013503, 8 pp., arXiv: 1703.04039 | DOI | MR | Zbl

[4] Alhaidari A. D., Open problem in orthogonal polynomials, arXiv: 1709.06081

[5] Alhaidari A. D., Quantum mechanics with orthogonal polynomials, arXiv: 1709.07652

[6] Alhaidari A. D., Bahlouli H., “Extending the class of solvable potentials. I The infinite potential well with a sinusoidal bottom”, J. Math. Phys., 49 (2008), 082102, 13 pp. | DOI | MR | Zbl

[7] Alhaidari A. D., Ismail M. E. H., “Quantum mechanics without potential function”, J. Math. Phys., 56 (2015), 072107, 19 pp., arXiv: 1408.4003 | DOI | MR | Zbl

[8] Chihara T. S., An introduction to orthogonal polynomials, Mathematics and its Applications, 13, Gordon and Breach Science Publishers, New York–London–Paris, 1978 | MR | Zbl

[9] Koekoek R., Lesky P. A., Swarttouw R. F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[10] Koepf W., Schmersau D., “Recurrence equations and their classical orthogonal polynomial solutions”, Appl. Math. Comput., 128 (2002), 303–327 | DOI | MR | Zbl

[11] Li Y., Private communication with Professor Alhaidari

[12] Olver F. W. J., Lozier D. W., Boisvert R. F., Clark C. W. (eds.), NIST handbook of mathematical functions, Cambridge University Press, Cambridge, 2010 | MR | Zbl

[13] Olver F. W. J., Olde Daalhuis A. B., Lozier D. W., Schneider B. I., Boisvert R. F., Clark C. W., Miller B. R., Saunders B. V. (eds.), NIST digital library of mathematical functions, Release 1.0.21 of 2018-12-15, http://dlmf.nist.gov

[14] Tcheutia D. D., Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or $q$-quadratic lattice, arXiv: 1901.03672

[15] Van Assche W., “Asymptotics for orthogonal polynomials and three-term recurrences”, Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 294, Kluwer Acad. Publ., Dordrecht, 1990, 435–462 | DOI | MR

[16] Van Assche W., “Compact Jacobi matrices: from Stieltjes to Krein and $M(a,b)$”, Ann. Fac. Sci. Toulouse Math., 1996, 195–215, arXiv: math.CA/9510214 | DOI | MR | Zbl