q-Hardy–Berndt type sums associated with q-Genocchi type zeta and q-l-functions

Abstract

a r t i c l e i n f o MSC: primary 11F20 11S80 secondary 30B40 11B68 11S40 30B50 44A05Keywords: q-Bernoulli numbers and polynomials q-Genocchi zeta q-l series p-adic interpolation function q-Dedekind type sums q-Hardy-Berndt type sums a b s t r a c tThe aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define qanalogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C p . Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and qHardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.

Special functions related to Dedekind-type DC-sums and their applications

“…By using p-adic q-Volkenborn integral, in [27] and [28], Kim defined p-adic q-Dedekind sums. In [49], [48], [52], [53], we defined q-Dedekind type sums, q-Hardy-Berndt type sums and p-adic q-Dedekind sums. By using same method, p-adic q-analogue of the sum T m (h, k) may be defined.…”

“…Some of them are given by 1, − 1 2 , 0, 1 4 , · · · , E n = 2 n E n ( 1 2 ) and E 2n = 0, (n ∈ N) cf. ( [28]- [37], [25], [39], [47], [53], [51]) and see also the references cited in each of these earlier works.…”

On generalized Dedekind sums involving quasi-periodic Euler functions

“…Dedekind [13] introduced the sum (1.1) in connection with the modular properties of the Dedekind η-function and deduced from his reciprocity law 12hk s(h, k) + s(k, h) = h 2 − 3hk + k 2 + 1 (1.3) (see [2, p. 62] and [19, p. 148]). In later years several mathematicians generalized s(h, k) and showed that the generalized functions too satisfy a reciprocity law, see [1][2][3][4][5][7][8][9][10][11][12]14,16,18,[20][21][22][23] and the references given there. The first proof of (1.3) does not employ the theory of the Dedekind η-function is due to Rademacher [17].…”

“…Can et al [8] gave twisted versions of the Frobenius-Euler polynomials, Dedekind type sums related to the Frobenius-Euler functions, and corresponding reciprocity law. In [20], Simsek constructed generating functions of q-Hardy-Berndt type sums and q-HardyBerndt type sums attached to Dirichlet character. He gave some relations and related to these sums.…”

A note on degenerate Genocchi and poly-Genocchi numbers and polynomials

“…When α = 1, G n = G (1) n (0) are called the Genocchi numbers. The reader may refer to [31] as an important work focusing on the Genocchi polynomials and their relations with different areas.…”