Catalan Numbers in Terms of $(\alpha, k)-$Gamma Function and $(\alpha, k)-$Beta Function

Year 2023, , 164 - 170, 30.06.2023

Abdullah Akkurt , Huseyin Yıldırım

https://doi.org/10.47000/tjmcs.1182387

Abstract

In the paper, the authors discuss some extended results involving the Catalan numbers and establish an integral representation of the Catalan numbers in terms of the $(\alpha,k)$-gamma and $(\alpha,k)$-beta function. We refer to the results available in the literature by giving special values to the parameters in the obtained theorems.

Keywords

Catalan number, $(\alpha,k)$-gamma function, $(\alpha,k)$-beta function

References

• Abdeljawad, T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279(2015), 57–66.
• Andrews, G.E., Askey, R., Roy, R., Encyclopedia of Mathematics and its Applications. Special Functions, 1999.
• Comtet, L., Advanced Combinatorics: The Art of Finite and Infinite Expansions, Springer Science and Business Media, 1974.
• Dana-Picard, T., Parametric integrals and Catalan numbers, International Journal of Mathematical Education in Science and Technology, 36(4)(2005), 410–414.
• Iyiola, O.S., Nwaeze, E.R., Some new results on the new conformable fractional calculus with application using D’Alambert approach, Progr. Fract. Differ. Appl, 2(2)(2016), 115–122.
• Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264(2014), 65–70.
• Koshy, T., Catalan Numbers with Applications, Oxford University Press, 2008.
• Lebedev, N.N., Silverman, R.A., Livhtenberg, D.B., Special functions and their applications, Physics Today, 18(12)(1965), 70.
• Qi, F., Parametric integrals, the Catalan numbers, and the beta function, Elemente der Mathematik, 72(3)(2017), 103–110.
• Qi, F., Akkurt, A., Yildirim, H., Catalan numbers, k-gamma and k-beta functions, and parametric integrals, J. Comput. Anal. Appl, 25(6)(2018), 1036–1042.
• Sarıkaya, M.Z., Akkurt, A., Budak, H., T¨urkay, M. E.,Yildirim, H., On some special functions for conformable fractional integrals, Konuralp Journal of Mathematics , 8(2)(2020), 376–383 .
• Shi, X.T., Liu, F. F., Qi, F., An integral representation of the Catalan numbers, Glob. J. Math. Anal, 3(3)(2015), 130–133.