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

SIGMA 12 (2016), 048, 14 pages      arXiv:1602.02724      https://doi.org/10.3842/SIGMA.2016.048
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases

Luc Vinet a and Alexei Zhedanov b
a) Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7 Canada
b) Institute for Physics and Technology, 83114 Donetsk, Ukraine

Received February 08, 2016, in final form May 07, 2016; Published online May 14, 2016; Reference [17] added May 22, 2016

Abstract
We introduce the notion of ''hypergeometric'' polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis $\varphi_n(x)$: $L \varphi_n(x) = \lambda_n \varphi_n(x) + \tau_n(x) \varphi_{n-1}(x)$ with some coefficients $\lambda_n$, $\tau_n$. We find the necessary and sufficient conditions for the polynomials $P_n(x)$ to be orthogonal. For the special cases where the sets $\lambda_n$ correspond to the classical grids, we find the complete solution to these conditions and observe that it leads to the most general Askey-Wilson polynomials and their special and degenerate classes.

Key words: abstract hypergeometric operator; orthogonal polynomials; classical orthogonal polynomials.

pdf (353 kb)   tex (18 kb)       [previous version:  pdf (352 kb)   tex (18 kb)]

References

1. Bannai E., Ito T., Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984, association schemes.
2. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
3. Durán A.J., Orthogonal polynomials satisfying higher-order difference equations, Constr. Approx. 36 (2012), 459-486.
4. Everitt W.N., Kwon K.H., Littlejohn L.L., Wellman R., Orthogonal polynomial solutions of linear ordinary differential equations, J. Comput. Appl. Math. 133 (2001), 85-109.
5. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
6. Geronimus J., The orthogonality of some systems of polynomials, Duke Math. J. 14 (1947), 503-510.
7. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
8. Leonard D.A., Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982), 656-663.
9. Okounkov A., On Newton interpolation of symmetric functions: a characterization of interpolation Macdonald polynomials, Adv. in Appl. Math. 20 (1998), 395-428.
10. Rains E.M., ${\rm BC}_n$-symmetric polynomials, Transform. Groups 10 (2005), 63-132, math.QA/0112035.
11. Roman S., The theory of the umbral calculus. I, J. Math. Anal. Appl. 87 (1982), 58-115.
12. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), 149-203, math.RA/0406555.
13. Terwilliger P., Leonard pairs from 24 points of view, Rocky Mountain J. Math. 32 (2002), 827-888, math.RA/0406577.
14. Terwilliger P., Leonard pairs and the $q$-Racah polynomials, Linear Algebra Appl. 387 (2004), 235-276, math.QA/0306301.
15. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form, J. Algebra 291 (2005), 1-45, math.RA/0304077.
16. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition, J. Comput. Appl. Math. 178 (2005), 437-452, math.RA/0306290.
17. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array, Des. Codes Cryptogr. 34 (2005), 307-332, math.RA/0306291.
18. Tsujimoto S., Vinet L., Zhedanov A., Dunkl shift operators and Bannai-Ito polynomials, Adv. Math. 229 (2012), 2123-2158, arXiv:1106.3512.
19. Vinet L., Zhedanov A., A 'missing' family of classical orthogonal polynomials, J. Phys. A: Math. Theor. 44 (2011), 085201, 16 pages, arXiv:1011.1669.
20. Zhedanov A., Abstract ''hypergeometric'' orthogonal polynomials, arXiv:1401.6754.