Related papers: Rademacher-Carlitz Polynomials
The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli…
In this article, we consider the reciprocal antisymmetric polynomial \[P(z) = \sum_{j = 0}^{s}(-1)^j\gamma_j\left(z^j - z^{N + s + 1 - j}\right), \ \gamma_0 = 1.\] It is shown that if all the zeros of $P(z)$ are located on the unit circle,…
Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a…
We prove a version of Gauss's Lemma. It recursively constructs polynomials {c_k} for k=0,1,...,m+n, in Z[a_i,A_i,b_j,B_j] for i=0,...,m, and j=0,1,...,n, having degree at most (m+n choose m) in each of the four variable sets, such that…
Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i…
We introduce the subsum polynomial of a partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_k)$ defined by $\mathrm{sp}(\lambda, x)=\prod_{i=1}^k(1+x^{\lambda_i})$. We study the sum of reciprocals of $\mathrm{sp}(\lambda, x)$ over all…
Let $n$ be any nonnegative integer and \[ D_n^{(h)}(x)=\sum_{k=0}^{n}\binom{n+k}{2k}^{h}\binom{2k}{k}^{h}{x}^{k} \text{ and } S_{n}^{(h)}(x)=\sum_{k=0}^{n}\binom{n+k}{2k}^{h}C_{k}^{h}{x}^{k} \] be the generalized Delannoy polynomials and…
Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \[S_r(n) = \sum_k \binom{2n}{k}|n-k|^r,\] where $r$ and $n$ are non-negative integers. We consider sums of the form…
Dedekind sums, arithmetic correlation sums that arose in Dedekind's study of the modular transformation of the logarithm of the eta-function, are surprisingly ubiquitous. Their arithmetic properties attracted the attention of number…
Recent work of Bettin and Conrey on the period functions of Eisenstein series naturally gave rise to the Dedekind-like sum \[ c_{a}\left(\frac{h}{k}\right) \ = \ k^{a}\sum_{m=1}^{k-1}\cot\left(\frac{\pi…
In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using…
Let $s(a,b)$ denote the classical Dedekind sum and $S(a,b)=12s(a,b)$. Let $k/q$, $q\in \Bbb N$, $k\in \Bbb Z$, $(k,q)=1$, be the value of $S(a,b)$. In a previous paper we showed that there are pairs $(a_r,b_r)$, $r\in\Bbb N$, such that…
We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial…
Vector-valued Jack polynomials associated to the symmetric group ${\mathfrak S}_N$ are polynomials with multiplicities in an irreducible module of ${\mathfrak S}_N$ and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators…
For a positive integer k and an arbitrary integer h, the Dedekind sum s(h,k) was first studied by Dedekind because of the prominent role it plays in the transformation theory of the Dedekind eta-function, which is a modular form of weight…
A natural question about Dedekind sums is to find conditions on the integers $a_1, a_2$, and $b$ such that $s(a_1,b) = s(a_2, b)$. We prove that if the former equality holds then $ b \ | \ (a_1a_2-1)(a_1-a_2)$. Surprisingly, to the best of…
In the present paper, for $s,t,u \in {\mathbb{C}}$, we show rapidly (or globally) convergent series representations of the Tornheim double zeta function $T(s,t,u)$ and (desingularized) symmetric Tornheim double zeta functions. As a…
Fixing a positive integer $r$ and $0 \le k \le r-1$, define $f^{\langle r,k \rangle}$ for every formal power series $f$ as $ f(x) = f^{\langle r,0 \rangle}(x^r)+xf^{\langle r,1 \rangle}(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle}(x^r).$…
We consider the sums $S(k)=\sum_{n=0}^{\infty}\frac{(-1)^{nk}}{(2n+1)^k}$ and $\zeta(2k)=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}$ with $k$ being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we…
In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For…