Some Finite Summation Identities Comprising Binomial Coefficients for Integrals of the Bernstein Polynomials and Their Applications

Year 2024, Volume: 11 Issue: 1, 156 - 163, 28.03.2024

Buket Şimşek

https://doi.org/10.54287/gujsa.1436339

Abstract

Certain finite sums, including the Catalan numbers, factorial functions, binomial coefficients, and their computational formulas are of indispensable importance both in probability and statistics applications and in other branches of science. The primary aim of this article is to give the integral representation of the finite sum containing the products of the Bernstein polynomials, given in our article, by applying the Beta function and the Euler gamma functions. Other aims of this paper are to bring to light novel finite sum formulae containing binomial coefficients by analyzing and unifying this integral representation. Finally, some relations among these sums, binomial coefficients, and the Catalan numbers are given. We also give the Wolfram language codes. By applying these codes to the finite sums, we give some numerical values.

Keywords

Finite Sum, Bernstein Polynomials, Beta Function, Euler Gamma Functions, Binomial Coefficients, Catalan Numbers

References

• Acikgoz, M., & Araci, S. (2010). On generating function of the Bernstein polynomials. AIP Conference Proceedings, 1281, 1141-1143. https://doi.org/10.1063/1.3497855
• Bernstein, S.N. (1912). Démonstration du theoreme de Weierstrass fondee sur la calcul des probabilites. Communications of the Kharkov Mathematical Society, 13, 1-2.
• Chattamvelli, R. & Shanmugam, R. (2020). Discrete distributions in engineering and the applied sciences. Morgan & Claypool Publishers Series. https://doi.org/10.1007/978-3-031-02425-2
• Gradshteyn, I. S., & Ryzhik, I. M. (2007). Table of integrals Series and Products (Seventh Edition). Academic Press is an imprint of Elsevier. https://doi.org/10.1016/C2009-0-22516-5
• Kaur, H., & Shrivastav, A. K. (2020). Summation formulae involving basic hypergeometric and truncated basic hypergeometric functions. Journal of Information and Computational Science, 1(4), 456-461. https://doi.org/10.15864/jmscm.1404
• Kelly, E. J. (1981). Finite-sum expressions for signal detection probabilities. Technical Report Massachusetts Institute of Technology Lincoln Laboratory.
• Kilar, N. (2023). A New Class of Generalized Fubini Polynomials and Their Computational Algorithms. Applicable Analysis and Discrete Mathematics, 17, 496–524. https://doi.org/10.2298/AADM210708023K
• Koshy, T. (2008). Catalan numbers with applications. Oxford University Press, New York.
• Kucukoglu, I. (2023). Identities for the multiparametric higher-order Hermite-based Peters-type Simsek polynomials of the first kind. Montes Taurus Journal of Pure and Applied Mathematics, 5(1), 102–123.
• Lorentz, G. G. (1986). Bernstein polynomials. Chelsea Publication Company, New York.
• Moll, V. H. (2014). Special integrals of Gradshteyn and Ryzhik the proofs (Volume 1). CRC Press, USA.
• Stanley, R. P. (2015). Catalan numbers. New York: Cambridge University Press. https://doi.org/10.1017/CBO9781139871495
• Stanley, R. P. (2021). Enumerative and algebraic combinatorics in the 1960's and 1970's. https://doi.org/10.48550/arXiv.2105.07884
• Simsek, B., & Yardimci, A. (2016). Using Bezier curves in medical applications. Filomat, 30(4), 937-943. https://doi.org/10.2298/FIL1604937S
• Simsek, B. (2019). Formulas derived from moment generating functions and Bernstein polynomials. Applicable Analysis and Discrete Mathematics, 13(3), 839-848. https://doi.org/10.2298/AADM191227036S
• Simsek, B. (2020). A note on characteristic function for Bernstein polynomials involving special numbers and polynomials. Filomat, 34(2), 543-549. https://doi.org/10.2298/FIL2002543S
• Simsek, Y. (2014). Generating functions for the Bernstein type polynomials: A new approach to deriving identities and applications for the polynomials. Hacettepe Journal of Mathematics and Statistics, 43(1), 1-14.
• Simsek, Y. (2015). Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers, Mathematical Methods in the Applied Sciences, 38, 3007-3021. https://doi.org/10.1002/mma.3276
• Srivastava, H. M., & Choi, J. (2012). Zeta and q-Zeta functions and associated series and integrals. Amsterdam, London and New York: Elsevier.
• Yalcin, F., & Simsek, Y. (2022). Formulas for characteristic function and moment generating functions of beta type distribution. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, RACSAM, 116(86). https://doi.org/10.1007/s13398-022-01229-1