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

#### Yılmaz Şimşek [1]

##### 28 221

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.
Combinatorial sums, Binomial identities, Generating functions, Functional equations, Beta polynomials, Beta function, Gamma function, Bernstein basis functions, Catalan numbers, Harmonic numbers, Young inequality
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Primary Language en Mathematics Mathematics Author: Yılmaz Şimşek (Primary Author)Institution: DEPARTMENT OF MATHEMATICSCountry: Turkey
 Bibtex @research article { hujms471066, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {Hacettepe University}, year = {2018}, volume = {47}, pages = {1144 - 1155}, doi = {}, title = {Combinatorial sums and binomial identities associated with the Beta-type polynomials}, key = {cite}, author = {Şimşek, Yılmaz} } APA Şimşek, Y . (2018). Combinatorial sums and binomial identities associated with the Beta-type polynomials. Hacettepe Journal of Mathematics and Statistics, 47 (5), 1144-1155. Retrieved from http://dergipark.gov.tr/hujms/issue/39860/471066 MLA Şimşek, Y . "Combinatorial sums and binomial identities associated with the Beta-type polynomials". Hacettepe Journal of Mathematics and Statistics 47 (2018): 1144-1155 Chicago Şimşek, Y . "Combinatorial sums and binomial identities associated with the Beta-type polynomials". Hacettepe Journal of Mathematics and Statistics 47 (2018): 1144-1155 RIS TY - JOUR T1 - Combinatorial sums and binomial identities associated with the Beta-type polynomials AU - Yılmaz Şimşek Y1 - 2018 PY - 2018 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1144 EP - 1155 VL - 47 IS - 5 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2017 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Combinatorial sums and binomial identities associated with the Beta-type polynomials %A Yılmaz Şimşek %T Combinatorial sums and binomial identities associated with the Beta-type polynomials %D 2018 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 47 %N 5 %R %U ISNAD Şimşek, Yılmaz . "Combinatorial sums and binomial identities associated with the Beta-type polynomials". Hacettepe Journal of Mathematics and Statistics 47 / 5 (October 2018): 1144-1155.