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

SIGMA 2 (2006), 034, 8 pages      math.CA/0603408      https://doi.org/10.3842/SIGMA.2006.034

On Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials

Valentyna A. Groza a and Ivan I. Kachuryk b
a) National Aviation University, 1 Komarov Ave., Kyiv, 03058 Ukraine
b) Khmel'nyts'kyi National University, Khmel'nyts'kyi, Ukraine

Received February 14, 2006, in final form February 28, 2006; Published online March 16, 2006

Abstract
The dual discrete q-ultraspherical polynomials Dn(s)(μ(x;s)|q) correspond to indeterminate moment problem and, therefore, have one-parameter family of extremal orthogonality relations. It is shown that special cases of dual discrete q-ultraspherical polynomials Dn(s)(μ(x;s)|q), when s = q-1 and s = q, are directly connected with q-1-Hermite polynomials. These connections are given in an explicit form. Using these relations, all extremal orthogonality relations for these special cases of polynomials Dn(s)(μ(x;s)|q) are found.

Key words: q-orthogonal polynomials; dual discrete q-ultraspherical polynomials; q-1-Hermite polynomials; orthogonality relation.

pdf (196 kb)   ps (145 kb)   tex (9 kb)

References

1. Askey R., Wilson J., Some basic hypergeometric polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc., 1985, V.319, 1-115.
2. Askey R., Wilson J., A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols, SIAM J. Math. Anal., 1979, V.10, 1008-1016.
3. Klimyk A., Spectra of observables in the q-oscillator and q-analogue of the Fourier transform, SIGMA, 2005, V.1, Paper 008, 17 pages, math-ph/0508032.
4. Askey R., Continuous q-Hermite polynomials when q > 1, in q-Series and Partitions, Editor D. Stanton, Berlin, Springer, 1998, 151-158.
5. Atakishiyev N.M., Klimyk A.U., On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl., 2005, V.306, 637-645, math.CA/0403159.
6. Ismail M.E.R., Masson D.R., q-Hermite polynomials, biorthogonal functions, and q-beta integrals, Trans. Amer. Math. Soc., 1994, V.346, 63-116.
7. Groza V., Representations of the quantum algebra suq(1,1) and discrete q-ultraspherical polynomials, SIGMA, 2005, V.1, Paper 016, 7 pages, math.QA/0511632.
8. Gasper G., Rahman M., Basic hypergeometric functions, Cambridge, Cambridge University Press, 1990.
9. Atakishiyev N.M., Klimyk A.U., Duality of q-polynomials, orthogonal on countable sets of points, Elect. Trans. Numer. Anal., to appear, math.CA/0411249.
10. Atakishiyev N.M., Klimyk A.U., On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials, J. Math. Anal. Appl., 2004, V.294, 246-257, math.CA/0307250.
11. Shohat J., Tamarkin J.D., The problem of moments, Providence, R.I., American Mathematical Society, 1943.