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Journals / Hacettepe Journal of Mathematics and Statistics / 2018 / Cilt: 47 - Sayı: 5

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

Yılmaz ŞİMŞEK
Pages
1144–1155
Publication Date
2022-10-18
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We also give another inequality for the Catalan numbers.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we first provide some functional equations of the generating functions for beta-type polynomials.
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Key Findings & Data:Using these equations, we derive various identities of the beta-type polynomials and the Bernstein basis functions.
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Conclusion & Impact:We also give another inequality for the Catalan numbers.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Combinatorial sums · Binomial identities · Generating functions

Abstract

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.

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Cite This Article

ŞİMŞEK, Y. (2022). Combinatorial sums and binomial identities associated with the Beta-type polynomials. Hacettepe Journal of Mathematics and Statistics, Cilt: 47 - Sayı: 5, 1144–1155. https://ajindex.com/articles/529840

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