Related papers: Higher Hickerson formula

We derive a formula for the evaluation of weighted generalized Fibonacci sums of the type $S_k^n (w,r) = \sum_{j = 0}^k {w^j j^r G_j{}^n }$. Several explicit evaluations are presented as examples.

We present an algorithm for computing all the solutions in not necessarily distinct integers to the decomposition of the unit into a sum of unit fractions with denominators $p^a.q^b$ where $p$ and $q$ are two distinct primes, each appearing…

For $a\in \Bbb Z$ and $b\in\Bbb N$, $(a,b)=1$, let $s(a,b)$ denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs $(a_i,b_i)$, $i\in\Bbb N$, with $b_i$…

We prove the Kloosterman-Spectral sum formula for PSL(2,Z[i])\PSL(2,C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. This sum formula allows…

The sum $S(h,k):=\sum_{j=1}^{k-1}(-1)^{j+1+[hj/k]}$ appears in the modular transformation formulae of the classical theta function $\vartheta_3(z)$. The double sum $S(k) := \sum_{h=1}^{k-1}S(h,k)$ has a remarkable distribution of values.…

Dedekind sums have applications in quite a number of fields of mathematics. Therefore, their distribution has found considerable interest. This article gives a survey of several aspects of the distribution of these sums. In particular, it…

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is…

We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results…

In a previous paper, I have defined non--commutative generalized Dedekind symbols for classical $PSL(2,Z)$--cusp forms using iterated period polynomials. Here I generalize this construction to forms of real weights using their iterated…

The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums, and give some exact computational formulae for them by using the properties of Gauss…

We introduce some new higher dimensional generalizations of the Dedekind sums associated with the Bernoulli functions and of those Hardy sums which are defined by the sawtooth function. We generalize a variant of Parseval's formula for the…

We introduce the inversion polynomial for Dedekind sums $f_b(x)=\sum x^{\operatorname{inv}(a,b)}$ to study the number of $s(a,b)$ which have the same value for given $b$. We prove several properties of this polynomial and present some…

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with…

In this paper we are concerned with a family of sums involving the floor function. With $r$ a non negative integer and $n$ and $m$ positive integers we consider the sums…

The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel…

In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let $n>2$ and $q>0$ be integers with $2\mid n$ or $2\nmid q$. We show that…

Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke…

In this paper we establish the generalized Beukers integral $I_{m}(a_{1},...,a_{n})$ with some methods of partial fraction decomposition. Thus one obtains an explicit expression of the generalized Beukers integral. Further, we estimate the…

We connect Dedekind sums and some formulas for numerical semigroups.

Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…