Mümün Can

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

Abstract 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 integral ∫ 0 x B n 1 ( b 1 z + y 1 ) ⋯ B n r ( b r z + y r ) d z , where 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.

Character analogues of certain Hardy-Berndt sums

In this paper we consider transformation formulas for \[ B\left( z,s:\chi\right) =\sum\limits_{m=1}^{\infty}\sum\limits_{n=0} ^{\infty}\chi(m)\chi(2n+1)\left( 2n+1\right) ^{s-1}e^{\pi im(2n+1)z/k}. \] We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties of them. With the help of the character analogues of the Euler--Maclaurin summation formula we establish integral representations for the Hardy-Berndt character sums

Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

s_{3,p}\left( d,c:\chi\right)

Extended Bernoulli and Stirling matrices and related combinatorial identities

and

Character analogue of the Boole summation formula with applications

s_{4,p}\left( d,c:\chi\right)

Transformation formulas of a character analogue of \(\log \theta _{2}(z)\)

.

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

In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity theorems are proved for these Dedekind sums. Furthermore, as an application of the transformation formulae, relations between various infinite series and evaluations of several infinite series are deduced. Finally, we consider these sums for some special cases.

On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions

Abstract In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli polynomials and Stirling numbers of both kinds, which generalize various known identities. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.

Twisted Dedekind Type Sums Associated with Barnes' Type Multiple Frobenius-Euler l-Functions

In this paper, we present the character analogue of the Boole summation formula. Using this formula, an integral representation is derived for the alternating Dirichlet

Degenerate and character Dedekind sums

L-