Related papers: On Pellarin's $L$-series

We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…

This paper investigates the zero distribution of a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $1+ct+B(z)t^{2}+A(z)t^{3}$ where $c\in\mathbb{R}$ and $A(z)$, $B(z)$ are real linear…

We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties…

In this paper we consider Dirichlet series absolutely converging for $\sigma>1$ with an Euler product, natural bounds on the coefficients and satisfying orthogonality relations of Selberg type. Let $N\geq 1$, $F_1(s),...,F_N(s)$ be as above…

Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a…

Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to…

An analogue of Serre's theorem is established for finite dimensional simple Lie superalgebras, which describes presentations in terms of Chevalley generators and Serre type relations relative to all possible choices of Borel subalgebras.…

For $0<\alpha<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^{\alpha}(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is…

We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem we construct sums of exponential functions having only real zeros, we construct…

The main object of this paper is to construct a new generating function of the (q-) Bernstein type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and…

The main object of this paper is to investigate a new class of the generalized Hurwitz type poly-Bernoulli numbers and polynomials from which we derive some algorithms for evaluating the Hurwitz type poly-Bernoulli numbers and polynomials.…

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit…

In this paper, we prove some zero density theorems for certain families of Dirichlet $L$-functions. More specifically, the subjects of our interest are the collections of Dirichlet $L$-functions associated with characters to moduli from…

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…

A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet…

Let $a=(a_n)_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\sum_{n\ge 1} a_n/n^s$, and $N_a(\sigma_1, \sigma_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $\sigma_1…

In the research, with aid of the Fa\`a di Bruno formula, be virtue of several identities for the Bell polynomials of the second kind, with help of two combinatorial identities, by means of the (logarithmically) complete monotonicity of…

Using non-archimedean q-integrals on Zp defined in [15, 16], we define a new Changhee q-Euler polynomials and numbers which are different from those of Kim [7] and Carlitz [2]. We define generating functions of multiple q-Euler numbers and…

In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell polynomials in the restricted case where q is a positive integer greater than 1. The arguments of the complete Bell polynomials comprise the…