On generalized Dedekind sums involving quasi-periodic Euler functions

Kim, Min‐Soo; Son, Jin-Woo

doi:10.1016/j.jnt.2014.05.022

Cited by 10 publications

(4 citation statements)

References 15 publications

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“…for gcd (b, c) = 1. For several proofs of (1) see [22] (for recent studies on Dedekind sums the reader may consult to [12,13,15,16,24]). These sums were later generalized by various mathematicians and the corresponding reciprocity laws were obtained.…”

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confidence: 99%

On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

Dağli

Can

2015

Journal of Number Theory

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We give a simple proof for the reciprocity formulas of character Dedekind sums associated with two primitive characters, whose modulus need not to be the same, by utilizing the character analogue of the Euler-MacLaurin summation formula. Moreover, we extend known results on the integral of products of Bernoulli polynomials by considering the integralwhere b l (b l = 0) and y l (1 ≤ l ≤ r) are real numbers. As a consequence of this integral we establish a connection between the reciprocity relations of sums of products of Bernoulli polynomials and of the Dedekind sums.

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confidence: 99%

On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

Dağli

Can

2015

Journal of Number Theory

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“…It is worth mentioning that Formula (46) was also discovered by Bayad ([27] Equation (1.2.13)), and has been used by Kim and Son [34] and Hu, Kim, and Kim [35] to establish some reciprocity formulas for the generalized Dedekind sums involving quasi-periodic Euler functions.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions

2023

Symmetry

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In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known results, for example, Raabe’s multiplication formula and Hermite’s identity, are deduced as special cases.

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“…Remark 3.4 Kim and Son [11] proved the reciprocity formula for generalized Dedekind sums T r (c, d) as…”

Section: Remark 33mentioning

confidence: 99%

“…Combining ( 9) and (11) gives the reciprocity relation for sums of products of higher-order Euler polynomials.…”

Section: Euler Polynomialsmentioning

confidence: 99%

On the integral of products of higher-order Bernoulli and Euler polynomials

Dağli¹,

Can²

2017

Funct. Approx. Comment. Math.

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In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same way, similar relations are obtained for l higher-order Bernoulli polynomials and r higher-order Euler polynomials. Moreover, we establish the connection between the results and the generalized Dedekind sums and Hardy-Berndt sums. Finally, the Laplace transform of Euler polynomials is given.

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On generalized Dedekind sums involving quasi-periodic Euler functions

Cited by 10 publications

References 15 publications

On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions

On the integral of products of higher-order Bernoulli and Euler polynomials

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