Identities on poly-Dedekind sums

Abstract

Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

Reciprocity of poly-Dedekind-type DC sums involving poly-Euler functions

“…The were considered and shown to satisfy a reciprocity relation in [16], where B were introduced and shown to satisfy a reciprocity relation in [13], where E p (x) is the pth Euler function. Simsek found trigonometric representations of the Dedekind-type DC sums and their relations to Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), and Hardy-Berndt sums.…”

“…were considered, and a reciprocity law for those sums was shown in [16,19]. Here B (k) p (x) are the type 2 poly-Bernoulli polynomials of index k, B (k) p (x) = B (k) p (x -[x]) (see [16]), and B (1) p (x) = B p (x). The Dedekind-type DC sums (see (8)) were first introduced and shown to satisfy a reciprocity relation in [13].…”

“…Komatsu [18] [10,15]) and also in constructing poly-Dedekind sums associated with such polynomials (see [16,19]). Here we recall from [10] that the type 2…”

“…Note that Li k (x) has order 1, so that the composition Li k (1e -t ) also has order 1. In addition, Li 1 (x) = -log(1x), so that B (1) n (x) are the ordinary Bernoulli polynomials with B (1) 1 (x) = x + 1 2 (see (16)). Thus we may say that Ei k (x) is a kind of a compositional inverse to Li k (x).…”

Two-Variable Type 2 Poly-Fubini Polynomials

“…Some mathematicians have considered and examined several extensions of special polynomials via polyexponential function, cf. [5,11,13,16,17] and see also the references cited therein. For example, Duran et al [11] defined type 2 poly-Frobenius-Genocchi polynomials by the following Maclaurin series expansion (in a suitable neighborhood of z = 0):…”