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

SIGMA 12 (2016), 071, 12 pages      arXiv:1607.06196      https://doi.org/10.3842/SIGMA.2016.071
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Report from the Open Problems Session at OPSFA13

Edited by Howard S. Cohl
Applied and Computational Mathematics Division, National Institute of Standards and Technology (NIST), Gaithersburg, MD, 20899-8910, USA

Received January 28, 2016, in final form July 12, 2016; Published online July 21, 2016

Abstract
These are the open problems presented at the 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA13), Gaithersburg, Maryland, on June 4, 2015.

Key words: Schur's inequality; hypergeometric functions; orthogonal polynomials; linearization coefficients; connection coefficients; symbolic summation; multiple summation; numerical algorithms; Gegenbauer polynomials; multiple zeta values; distribution of zeros.

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References

1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
2. Aptekarev A.I., Geronimo J.S., Measures for orthogonal polynomials with unbounded recurrence coefficients, J. Approx. Theory 207 (2016), 339-347, arXiv:1408.5349.
3. Askey R., Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975.
4. Bailey W.N., Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 32, Stechert-Hafner, Inc., New York, 1964.
5. Beatson R.K., zu Castell W., Xu Y., A Pólya criterion for (strict) positive-definiteness on the sphere, IMA J. Numer. Anal. 34 (2014), 550-568, arXiv:1110.2437.
6. Blumenthal O., Über die Entwicklung einer willkürlichen Funktion nach den Nennern des Kettenbruches $\int_{-\infty}^0(z-\xi)^{-1}\varphi(\xi)d\xi$, Dissertation, Georg-August-Universität Göttingen, Göttingen, Germany, 1898.
7. Chihara T.S., Orthogonal polynomials whose zeros are dense in intervals, J. Math. Anal. Appl. 24 (1968), 362-371.
8. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
9. Chihara T.S., Spectral properties of orthogonal polynomials on unbounded sets, Trans. Amer. Math. Soc. 270 (1982), 623-639.
10. Chihara T.S., The one-quarter class of orthogonal polynomials, Rocky Mountain J. Math. 21 (1991), 121-137.
11. Chihara T.S., An analog of the Blumenthal-Nevai theorem for unbounded intervals, J. Comput. Appl. Math. 153 (2003), 535-536.
12. Chyzak F., An extension of Zeilberger's fast algorithm to general holonomic functions, Discrete Math. 217 (2000), 115-134.
13. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491-1552.
14. Deift P.A., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Vol. 3, New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, RI, 1999.
15. Dunkl C.F., An intertwining operator for the group $B_2$, Glasg. Math. J. 49 (2007), 291-319, math.CA/0607823.
16. Dunkl C.F., Gasper G., The sums of a double hypergeometric series and of the first $m+1$ terms of ${}_3F_2(a,b,c;(a+b+1)/2,2c;1)$ when $c=-m$ is a negative integer, arXiv:1412.4022.
17. Dunkl C.F., Slater P.B., Separability probability formulas and their proofs for generalized two-qubit X-matrices endowed with Hilbert-Schmidt and induced measures, Random Matrices Theory Appl. 4 (2015), 1550018, 18 pages, arXiv:1501.02289.
18. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. (Editors), Higher transcendental functions, Vol. I, Bateman Manuscript Project, McGraw-Hill Book Co., New York, 1953.
19. Erdős L., Péché S., Ramírez J.A., Schlein B., Yau H.-T., Bulk universality for Wigner matrices, Comm. Pure Appl. Math. 63 (2010), 895-925, arXiv:0905.4176.
20. Hale N., Townsend A., Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights, SIAM J. Sci. Comput. 35 (2013), A652-A674.
21. Hale N., Townsend A., A fast, simple, and stable Chebyshev-Legendre transform using an asymptotic formula, SIAM J. Sci. Comput. 36 (2014), A148-A167.
22. Hardy G.H., Littlewood J.E., Pólya G., Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.
23. Hilbert D., Ueber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), 342-350.
24. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
25. Ismail M.E.H., Zeng J., A combinatorial approach to the 2D-Hermite and 2D-Laguerre polynomials, Adv. in Appl. Math. 64 (2015), 70-88.
26. Koutschan C., Advanced applications of the holonomic systems approach, Ph.D. Thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, 2009, available at http://www.risc.jku.at/research/combinat/software/HolonomicFunctions.
27. Kuijlaars A.B.J., McLaughlin K.T.-R., Van Assche W., Vanlessen M., The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$, Adv. Math. 188 (2004), 337-398, math.CA/0111252.
28. Miller Jr. W., Special functions and the complex Euclidean group in $3$-space. II, J. Math. Phys. 9 (1968), 1175-1187.
29. Nevai P.G., Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), v+185 pages.
30. NIST digital library of mathematical functions, Release 1.0.10 of 2015-08-07, online companion to [31], available at http://dlmf.nist.gov/.
31. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010, print companion to [30].
32. Rahman M., A nonnegative representation of the linearization coefficients of the product of Jacobi polynomials, Canad. J. Math. 33 (1981), 915-928.
33. Rahman M., The linearization of the product of continuous $q$-Jacobi polynomials, Canad. J. Math. 33 (1981), 961-987.
34. Sánchez-Ruiz J., Artés P.L., Martínez-Finkelshtein A., Dehesa J.S., General linearization formulae for products of continuous hypergeometric-type polynomials, J. Phys. A: Math. Gen. 32 (1999), 7345-7366.
35. Schneider C., Solving parameterized linear difference equations in terms of indefinite nested sums and products, J. Difference Equ. Appl. 11 (2005), 799-821.
36. Świderski G., Spectral properties of unbounded Jacobi matrices with almost monotonic weight, Constr. Approx. 44 (2016), 141-157, arXiv:1501.03420.
37. Takayama N., An algorithm of constructing the integral of a module - an infinite dimensional analog of Gröbner basis, in Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), ACM, New York, 1990, 206-211.
38. Tao T., Vu V., Random matrices_ universality of local eigenvalue statistics up to the edge, Comm. Math. Phys. 298 (2010), 549-572, arXiv:0908.1982.
39. Townsend A., Trogdon T., Olver S., Fast computation of Gauss quadrature nodes and weights on the whole real line, IMA J. Numer. Anal. 36 (2016), 337-358, arXiv:1410.5286.
40. Wilf H.S., Zeilberger D., An algorithmic proof theory for hypergeometric (ordinary and ''$q$'') multisum/integral identities, Invent. Math. 108 (1992), 575-633.
41. Zeilberger D., A fast algorithm for proving terminating hypergeometric identities, Discrete Math. 80 (1990), 207-211.
42. Zeilberger D., A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321-368.
43. Zudilin W., On a family of polynomials related to $\zeta(2,1)=\zeta(3)$, arXiv:1504.07696.