(q,h)-Bernstein Bases and Basic Hypergeometric Series

Abstract

Quantum (q,h)-Bernstein bases and basic hypergeometric series are two seemingly unrelated mathematical entities. In this work, it is indicated that they are deeply interrelated theories. This new insight into two theories enables the provision of new proofs for two basic hypergeometric sums. The q-Chu-Vandermonde formula for basic hypergeometric series is proved by the partition of unity property for (q,h)-Bernstein bases, and the q-Pffaf-Saalschütz formula for basic hypergeometric series is proved by the Marsden identity for (q,h)-Bernstein bases.

Keywords

• Quantum Bernstein bases
• Marsden identity
• basic hypergeometric series
• q-Chu-Vandermonde formula
• q-Pfaff-Saalschütz formula

References

1. Andrews, G., Askey, R., Roy, R., “Special Functions”, Cambridge University Press, Cambridge 71 (1999).
2. Gasper, G., Rahman, M., “Basic Hypergeometric Series”, Cambridge University Press, Cambridge 96 (2004).
3. Ismail, M.E.H., “Classical and Quantum Orthogonal Polynomials in One Variable”, Cambridge University Press, Cambridge 98 (2005).
4. Bailey, W.N., “Generalized hypergeometric series.” Cambridge: Cambridge University Press (1935).
5. Zürnacı, F., Goldman, R., Simeonov, P., “Relationships between identities for quantum Bernstein bases and formulas for hypergeometric series”, Filomat 34(8) (2020) : 2485-2494.
6. Goldman, R., Simeonov, P. “Quantum Bernstein bases and quantum Bézier curves “, Journal of Computational and Applied Mathematics 288 (2015) : 284-303.
7. Tuncer, O.O., Simenov, P., Goldman, R., “Basic hypergeometric formulas and identities for negative degree q-Bernstein bases”, Filomat 38(8) (2024) : 2941-2948.
8. Ismail, M.E.H., Simenov, P., “Formulas and identities for the Askey-Wilson operator”, Advances in Applied Mathematics 76 (2016) : 68-96.

9. Chu, W., “Divided differences and terminating well-poised hypergeometric series”, Integral Transforms and Special Functions 31(8) (2020) : 655-668.
10. Chu, W., “Divided differences and well-poised q-series”, Journal of Difference Equations and Applications 28(4) (2022) : 485-495.
11. Oruc, H., Phillips, G.M., “A generalization of the Bernstein polynomials”, Proceedings of the Edinburgh Mathematical Society 42 (1999) : 403-413.
12. Phillips, G.M., “A de Casteljau algorithm for generalized Bernstein polynomials”, BIT 37 (1997) : 232-236.
13. Phillips, G.M., “Bernstein polynomials based on the q-integers”, Annals of Numerical Mathematics 4 (1997) : 511-518.
14. Phillips, G.M., “A survey of results on the q-Bernstein polynomials”, IMA Journal of Numerical Analysis 30 (2010) : 277-288. Simeonov, P., Zafiris, V., Goldman, R., “h-Blossoming: A new approach to algorithms and identities for h-Bernstein bases and h-Bѐzier curves”, Computer Aided Geometric Design 28(9) (2011) : 549-565.
15. Simeonov, P., Zafiris, V., Goldman, R., “q-Blossoming: A new approach to algorithms and identities for q-Bernstein bases and q-Bѐzier curves”, Journal of Approximation Theory 164(1) (2012) : 77-104.
16. Stancu, D., “Approximation of functions by a new class of linear polynomial operators”, Revue Roumaine de Mathematiques Pures et Appliquees 13 (1968) : 1173-1194. Stancu, D., “Generalized Bernstein approximating operators. In: Itinerant Seminar on Functional Equations”, Approximation and Convexity,Cluj-Napoca (1984) : 185-192.