Combinatorial sums and binomial identities associated with the Beta-type polynomials

Yıl 2018, Cilt: 47 Sayı: 5, 1144 - 1155, 16.10.2018

Yılmaz Şimşek

Öz

In this paper, we first provide some functional equations of the generating functions for beta-type polynomials. Using these equations, we derive various identities of the beta-type polynomials and the Bernstein basis functions. We then obtain some novel combinatorial identities involving binomial coefficients and combinatorial sums. We also derive some generalizations of the combinatorics identities which are related to the Gould's identities and sum of binomial coefficients. Next, we present some remarks, comments, and formulas including the combinatorial identities, the Catalan numbers, and the harmonic numbers. Moreover, by applying the classical Young inequality, we derive a combinatorial inequality related to beta polynomials and combinatorial sums. We also give another inequality for the Catalan numbers.

Anahtar Kelimeler

Combinatorial sums, Binomial identities, Generating functions, Functional equations, Beta polynomials, Beta function, Gamma function, Bernstein basis functions, Catalan numbers, Harmonic numbers, Young inequality

Kaynakça

• Amdeberhan, T., Angelis, V. D., Lin, M., Moll, V. H. and Sury, B. A pretty binomial identity, Elem. Math. 2012 ;67: 18-25.
• Bajunaid, I, Cohen, J. M., Colonna, F. and Singman. D. Function Series, Catalan Numbers, and RandomWalks on Trees, Math. Association America. 112, 765-785, 2005.
• Bhandari, A. and Vignat, C., A probabilistic interpretation of the Volkenborn integral, arXiv:1201.3701v1.
• Choi, J. and Srivastava, H. M. Certain families of series associated with the Hurwitz-Lerch zeta function, Appl. Math. Comput. 170, 399-409, 2005.
• Chu, W. Summation formulae involving harmonic numbers , Filomat 26, 143-152, 2012.
• Clarrk, D. S. A class of combinatorial identities Discrete Appl. Math. 4, 325-327, 1982.
• Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, (Translated from the French by J. M. Nienhuys, Reidel, Dordrecht and Boston, 1974).
• Flajolet, P. and Sedgewick, R. Analytic Combinatorics, (Cambridge University Press, 2009).
• Goldman, R. Identities for the Univariate and Bivariate Bernstein Basis Functions. Graphics Gems V, (edited by Alan Paeth, Academic Press 1995; 149-162).
• Gould, H. W. Combinatorial Identities, (Morgantown Printing and Binding Co., Morgantown, WV, 1972).
• Gould, H. W. Combinatorial Identities, Vol.1-Vol.8, http://www.math.wvu.edu/~gould/
• Graham, R. L., Knuth, D. E. and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, (Second Edition, Addison-Wesley Publishing Company, Massachusetts, 1989).
• Hilton, P. and Persen, J. Catalan numbers, their generalization, and their uses, Math. Intelligencer, 13 (2), 64-75, 1991.
• Kim, T. A note on Catalan numbers associated with $p$-adic integral on $\mathbb{Z}_p$, https://arxiv.org/pdf/1606.00267v1.pdf
• Kittaneh, F. and Manasrah, Y. Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl. 361, 262-269, 2010.
• Mansour, T. Combinatorial identities and inverse binomial coefficients, Adv. in Appl. Math. 28, 196-202, 2002.
• Milovanovic, G. V. Extremal problems and inequalities of Markov-Bernstein type for polynomials, In Analytic and Geometric Inequalities and Applications, (Th.M. Rassias, H.M. Srivastava, eds.), Mathematics and Its Applications, 478, 245-264, Kluwer, Dordrecht 1999; MR1785873 (2001i:41013).
• Milovanovic, G. V. and, Rassias, Th. M. Inequalities for polynomial zeros, In survey on classical inequalities, (Th. M. Rassias, ed. Kluwer, Dordrecht, Math. Its Appl. 517, 165- 202, 2000).
• Rainville, E. D. Special functions (The Macmillan Company, New York, 1960).
• Simsek, Y. Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions, Fixed Point Theory Appl. 2003; 80, 2013, doi:10.1186/1687-1812-2013-80.
• Simsek, Y. $q$-Beta polynomials and their applications, Appl. Math. Inf. Sci. 7 (6), 2539-2547, 2013.
• Simsek, Y. A new class of polynomials associated with Bernstein and beta polynomials, Math. Meth. Appl. Sci. 37 (5), 676-685, 2014.
• Simsek, Y. Generating Functions for the Bernstein type polynomials: a new approach to deriving identities and applications for the polynomials, Hacet. J. Math. Stat. 43(1), 1-14, 2014.
• Simsek, Y. Beta-type polynomials and their generating functions, Appl. Math. Comput. 254, 172-182, 2015.
• Simsek, Y. A new combinatorial approach to analysis: Bernstein basis functions, combina- torial identities and Catalan numbers, Math. Meth. Appl. Sci. 38 (14), 3007-3021, 2015.
• Srivastava, H. M. Some generalizations of a combinatorial identities of L. Vietories, Discrete Math. 65, 99-102, 1987.
• Srivastava, H. M. and Choi, J. Series Associated with the Zeta and Related Functions (Kluwer Acedemic Publishers, Dordrecht, Boston and London, 2001).
• Sury, B. Sum of the reciprocals of the binomial coefficients, European J. Combin. 14, 351-353, 1993.
• Trif, T. Combinatorial sums and series involving inverses of binomial coefficients, Fibonacci Quart. 38, 79-84, 2000.
• https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds _and_asymptotic_formulas