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Journals / Turkish Journal of Mathematics / 2020 / Cilt: 44 - Sayı: 6

New classes of Catalan-type numbers and polynomials with their applications related to p-adic integrals and computational algorithms

Yılmaz ŞİMŞEK
Burçin ŞİMŞEK
İrem KÜÇÜKOĞLU
Pages
2337–2355
Publication Date
2022-11-01
DOI
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AI Summary & Key Takeaways

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

With the help of these integral formulas, we give some identities and relations associated with some classes of special numbers and also the Cauchy-type numbers.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The aim of this paper is to construct generating functions for new classes of Catalan-type numbers and polynomials.
📊
Key Findings & Data:By implementing these algorithms in the Python programming language, we illustrate the Catalan-type numbers and polynomials with their plots under the special conditions.
💡
Conclusion & Impact:With the help of these integral formulas, we give some identities and relations associated with some classes of special numbers and also the Cauchy-type numbers.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

The aim of this paper is to construct generating functions for new classes of Catalan-type numbers and polynomials. Using these functions and their functional equations, we give various new identities and relations involving these numbers and polynomials, the Bernoulli numbers and polynomials, the Stirling numbers of the second kind, the Catalan numbers and other classes of special numbers, polynomials and functions. Some infinite series representations, including the Catalan-type numbers and combinatorial numbers, are investigated. Moreover, some recurrence relations and computational algorithms for these numbers and polynomials are provided. By implementing these algorithms in the Python programming language, we illustrate the Catalan-type numbers and polynomials with their plots under the special conditions. We also give some derivative formulas for these polynomials. Applying the Riemann integral, contour integral, Volkenborn (bosonic p-adic) integral and fermionic p-adic integral to these polynomials, we also derive some integral formulas. With the help of these integral formulas, we give some identities and relations associated with some classes of special numbers and also the Cauchy-type numbers.

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Cite This Article

ŞİMŞEK, Y., ŞİMŞEK, B., KÜÇÜKOĞLU, İ. (2022). New classes of Catalan-type numbers and polynomials with their applications related to p-adic integrals and computational algorithms. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 6, 2337–2355. https://ajindex.com/articles/607494

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