相关论文: On a Class of Polynomials Generated by F (xt -- R(…

Let $\Phi$ be an irreducible (possibly noncrystallographic) root system of rank $l$ of type $P$. For the corresponding cluster complex $\Delta(P)$, which is known as pure $(l-1)$-dimensional simplicial complex, we define the generating…

Formal power series come up in several areas such as formal language theory , algebraic and enumerative combinatorics, semigroup theory, number theory etc. This paper focuses on the set x R[[x]] consisting of formal power series with zero…

For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$…

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of…

For a polynomial $f(t) = 1+f_0t+\cdots +f_{d-1}t^d$ with positive integer coefficients Bell and Skandera ask if real rootedness of f(t) implies that there is a simplicial complex with f-vector $(1,f_0 \ldots,f_{d-1})$. In this paper we…

We show that the only polynomial sets with a generating function of the form F (xt -- R(t)) and satisfying a three-term recursion relation are the monomial set and the rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first…

Let $P(x)$ be a polynomial of degree $m$, with nonnegative and non-decreasing coefficients. We settle the conjecture that for any positive real number $d$, the coefficients of $P(x+d)$ form a unimodal sequence, of which the special case $d$…

Let $F(t,u)\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F\_1, ..., F\_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$…

Let $(a_n)_{n \geq 0}$ be a sequence of integers such that its generating series satisfies $\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$ for some polynomial $h(t)$. For any $r \geq 1$ we study the coefficient sequence of the numerator…

The Brenke type generating functions are the polynomial generating functions of the form $$\sum_{n=0}^{\infty}{P_n(x )\over n!}t^n=A(t)B(xt), $$ where $A$ and $B$ are two formal power series subject to the conditions…

Let $A$ be a non-projectively-pluripolar set in a Fr\'{e}chet space $E.$ We give sufficient conditions to ensure the convergence on some zero-neighbourhood in $E$ of a (sequence of) formal power series of Fr\'{e}chet-valued continuous…

For a polynomial $P$ of degree greater than one, we show the existence of patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our…

In this paper we construct a generating polynomial over the rationals for the generic Newton polygon for the L function of exponential sums of the family of f = x^d+ a x^s parameterized by a, and prove some of its key properties. The…

The set ${\cal P}^{m\times n}_{r,d}$ of $m \times n$ complex matrix polynomials of grade $d$ and (normal) rank at most $r$ in a complex $(d+1)mn$ dimensional space is studied. For $r = 1, \dots , \min \{m, n\}-1$, we show that ${\cal…

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form $G(z,t)=P(t)+zt^{r}$, where the zeros of $P$ are positive and real. We show that every member of…

Fibonacci polynomials are generalizations of Fibonacci numbers, so it is natural to consider polynomial versions of the various results for Fibonacci numbers. According to Hong, Pongsriiam, Bulawa, and Lee, the generating function of the…

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which…

We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the…

Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable as Schur-Szeg\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give…

Considering the L-function of exponential sums associated to a polynomial over a finite field F_q, Deligne proved that a reciprocal root's p-adic order is a rational number in the interval [0, 1]. Based on hypergeometric theory, in this…