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When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume…

The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…

We prove polynomial boson-fermion identities for the generating function of the number of partitions of $n$ of the form $n=\sum_{j=1}^{L-1} j f_j$, with $f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of the…

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…

$T$-adic exponential sums associated to a Laurent polynomial $f$ are introduced. They interpolate all classical $p^m$-power order exponential sums associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of $T$-adic…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit…

In this paper, we introduce a certain method to construct polynomials producing many absolute pseudoprimes. By this method, we give new polynomials producing absolute pseudoprimes with any fixed number of prime factors which can be viewed…

We studies the Newton polygon for the L-function of toric exponential sums attached to a family of two variable generalized hyperkloosterman sum,$f_{t}(x,y)=x^{n}+y+\frac{t}{xy}$ with $t$ the parameter. The explicit Newton polygon is…

To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when…

We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In…

We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…

In this paper we obtain further improvement of index bounds for character sums of polynomials over finite fields. We present some examples, which show that our new bound is an improved bound compared to both the Weil bound and the index…

For the Schur polynomials bounded and unbounded generalizations of the Cauchy identities are found.

We prove effective finiteness results concerning polynomial values of the sums $$ b^k +\left(a+b\right)^k + \cdots + \left(a\left(x-1\right) + b\right)^k $$ and $$ b^k - \left(a+b\right)^k + \left(2a+b\right)^k - \ldots + (-1)^{x-1}…

The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…

We give an overview of classical summation formulations, such as Poisson's and Voronoi's, and then turn to modern versions involving modular form coefficients. A new formula involving the coefficients of cusp forms on GL(3) is described,…

We study arithmetic and asymptotic properties of polynomials provided by $Q_n(x):= x \sum_{k=1}^n k \, Q_{n-k}(x)$ with initial value $Q_0(x)=1$. The coefficients satisfy a central limit theorem and a local limit theorem involving Fibonacci…

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…

The aim of the present work is to show that recent results of the authors on the approximation of distributions of sums of independent summands by the infinitely divisible laws on convex polyhedra can be shown via an alternative class of…