Abstract
We derive in a unified way the difference equations for Askey–Wilson polynomials and their Stieltjes transforms, by using basic properties of the de Rham cohomology associated with q-integral representations (Jackson integrals of BC 1 type) of these functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBS Regional Conference Series, vol. 66. Am. Math. Soc., Providence (1986)
Aomoto, K.: q-analogue of de Rham cohomology associated with Jackson integrals. Proc. J. Acad. 66(7), 240–244 (1990)
Aomoto, K.: Connections formulas in the q-analog de Rham cohomology. In: Functional Analysis on the Eve of the 21 St Century, vol 1, New Brunswick, NJ, 1993, Prog. Math., vol. 131, pp. 1–12. Birkhauser, Boston (1995)
Andrews, G., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and Its Applications. University of Cambridge, Cambridge (1999)
Aomoto, K., Ito, M.: On the structure of Jackson integrals of BC n type. Tokyo J. Math. 31, 449–477 (2008)
Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (1985)
Brown, M., Evans, D., Ismail, M.: The Askey–Wilson polynomials and q-Sturm–Liouville problems. Math. Proc. Camb. Philos. Soc. 119, 1–16 (1996). Part 1
Ismail, M.: An operator calculus for the Askey–Wilson operator. Dedicated to the memory of G.C. Rota. Ann. Comb. 5, 347–362 (2001)
Ismail, M.: Lectures on q-orthogonal polynomials. Dept. of Math., Univ. of South Florida (2000)
Ito, M.: Symmetry classification for Jackson integrals associated with the root system BC n type. Compos. Math. 136, 209–216 (2003)
Ito, M.: Askey–Wilson type integrals associated with root systems. Ramanujan J. 12, 131–151 (2006)
Ito, M.: Convergence and asymptotic behaviour of Jackson integrals associated with irreducible reduced root systems. J. Approx. Theory 124, 154–180 (2003)
Rahman, M.: q-Wilson functions of the second kind. SIAM J. Math. Anal. 17, 1280–1286 (1986)
Szegö, G.: Orthogonal Polynomials. Am. Math. Soc., Providence (1975)
Aomoto, K.: A normal form of a holonomic q-difference system and its application to BC 1-type. Int. J. Pure. Appl. Math. 50, 85–95 (2009)
Ismail, M.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge Univ. Press, Cambridge (2005)
Odake, S., Sasaki, R.: Exactly solvable discrete quantum mechanics; shape invariance, Heisenberg solutions, annihilation–creation operators and coherent states. Prog. Theor. Phys. 119, 663–700 (2008)
Acknowledgements
The author is grateful to Prof. M. Ito for a stimulating discussion. The author appreciates some useful information provided by M. Ismail.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aomoto, K. Integral representations and holonomic q-difference structure of Askey–Wilson type functions. Ramanujan J 31, 239–255 (2013). https://doi.org/10.1007/s11139-012-9461-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-012-9461-z
Keywords
- Askey–Wilson type functions
- q-Integral representations
- Stieltjes transforms
- Holonomic q-difference equations