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Full-Text Articles in Physical Sciences and Mathematics
Higher-Order Character Dedekind Sum, Mümün Can
Higher-Order Character Dedekind Sum, Mümün Can
Turkish Journal of Mathematics
In this paper, we are interested in higher-order character Dedekind sum% \[ \sum\limits_{v=0}^{ck-1}\chi_{1}\left( v\right) \mathcal{B}_{p,\chi_{2}% }\left( a\frac{v+z}{c}+x\right) \mathcal{B}_{q}\left( b\frac{v+z}% {ck}+y\right) ,\text{ }a,b,c\in\mathbb{N} \text{ and }x,y,z\in\mathbb{R}, \] where $\chi_{1}$ and $\chi_{2}$ are primitive characters of modulus $k,$ $\mathcal{B}_{p}\left( x\right) $ and $\mathcal{B}_{p,\chi_{2}}\left( x\right) $ are Bernoulli and generalized Bernoulli functions, respectively. We employ the Fourier series technique to demonstrate reciprocity formulas for this sum. Derived formulas are analogues of Mikolas' reciprocity formula. Moreover, we offer Petersson--Knopp type identities for this sum.
New Classes Of Catalan-Type Numbers And Polynomials With Their Applications Related To $P$-Adic Integrals And Computational Algorithms, İrem Küçükoğlu, Burçi̇n Şi̇mşek, Yilmaz Şi̇mşek
New Classes Of Catalan-Type Numbers And Polynomials With Their Applications Related To $P$-Adic Integrals And Computational Algorithms, İrem Küçükoğlu, Burçi̇n Şi̇mşek, Yilmaz Şi̇mşek
Turkish Journal of Mathematics
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 …