A note on hardy type sums and Dedekind sums

Abstract

In [9], Cetin et al. defined a new special finite sum which is denoted by C1(h,k). In this paper, with the help of two-term polynomial relation, we will give the explicit values of the sum C1(h,k). We will see that for the odd values of h and k, this sum only depends on one variable. After that we will give many properties of this sum and connections with other well-known finite sums such as the Dedekind sums, the Hardy sums and the Simsek sums Y(h,k). By using the Fibonacci numbers and two-term polynomial relation, we will also give some relations for these sums.

Analytic Properties of the Sum B1(h, k)

“…From [10], we know that equation (29) holds true. If we also use ( 9), then we get the desired result.…”

“…For the odd values of k, the below theorem is given in [10]: Theorem 1.9. If (h, k) = 1, h and k are odd integers with k > 0, then we have…”

Identities for a Special Finite Sum Related to the Dedekind Sums and Fibonacci Numbers

“…The author (Cetin, 2016a) gave the following formula involving a relation between the Simsek sum 𝑌(𝜇, 𝛽) and the Fibonacci numbers:…”

“…This has many relations with other well known finite sums. Cetin (2016a;2016b) gave some useful properties for this sum.…”

Some Identities Involving Certain Hardy Sums and General Kloosterman Sums

“…The sums S 1 (h, k) are sometimes called Hardy sums. Some authors studied the properties of S 1 (h, k) and related sums, and obtained some interesting results, see [9][10][11][12]. A relation between certain Hardy sums S 1 (h, k) and classical Dedekind sums S(h, k) can be obtained in [12] that if (h, k) = 1, then…”