Related papers: Asymptotic estimate for the polynomial coefficient…

The weighted binomial sum $f_m(r)=2^{-r}\sum_{i=0}^r\binom{m}{i}$ arises in coding theory and information theory. We prove that,for $m\not \in\{0,3,6,9,12\}$, the maximum value of $f_m(r)$ with $0\leqslant r\leqslant m$ occurs when…

Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to…

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central…

Let $r,n$ be two natural numbers and let $H(r,n)$ denote the maximal absolute value of $r$th coefficient of divisors of $x^n-1$. In this paper, we show that $\sum_{n\leq x}H(r,n)$ is asymptotically equal to $c(r)x(\log x)^{2^r-1}$ for some…

Recently, Glasby and Paseman considered the following sequence of binomial sums $\{2^{-r}\sum_{i=0}^{r}\binom{m}{i}\}_{r=0}^{m}$ and showed that this sequence is unimodal and attains its maximum value at $r=\lfloor\frac{m}{3}\rfloor+1$ for…

Using well-known facts on Jacobi polynomials, we derive some asymptotic estimates for the maximum absolute value of generalized Gegenbauer polynomials.

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by…

If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to…

Let k and n be positive integers. We mainly show that $$(ln+1) | k\binom{kn+ln}{kn},$$ $$2\binom{kn}n | \binom {2n}{n}C_{2n}^{(k-1)}$$, $$\binom{kn}n | (2k-1)C_n\binom{2kn}{2n},$$ $$\binom{2n}n | (k+1)C_n^{(k-1)}\binom{2kn}{kn},$$…

We study the number of prime polynomials of degree $n$ over $\mathbb{F}_q$ in which the $i^{th}$ coefficient is either preassigned to be $a_i \in \mathbb{F}_q$ or outside a small set $S_i \subset \mathbb{F}_q$. This serves as a function…

It is shown how to obtain an asymptotic expansion of the generalised central trinomial coefficient $[x^n](x^2 + bx + c)^n$ by means of singularity analysis, thus proving a conjecture of Zhi-Wei Sun.

Let $S$ be a finite set of pairwise coprime positive integers and $Ax^2+Bx$ be an integer valued polynomial with $A> B\ge 0$. For integers $k\ge 1$ and $n\ge 0$, the coefficients $\gamma_{S,A,B}^k (n)$ are defined as \begin{align*}…

We propose a formula for finding the horizontal, oblique or curvilinear asymptote of any rational polynomial function of any positive degree, as a sum of matrix determinants formed directly from the coefficients of the terms in the given…

For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we…

In previous work we computed the number $C_n(q)$ of ideals of codimension $n$ of the algebra ${\mathbb{F}}_q[x,y,x^{-1}, y^{-1}]$ of two-variable Laurent polynomials over a finite field: it turned out that $C_n(q)$ is a palindromic…

Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size…

For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence…

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq…

In this article, we offer group-theoretic, field-theoretic, and topological interpretations of the Gaussian binomial coefficients and their sum. For a finite $p$-group $G$ of rank $n$, we show that the Gaussian binomial coefficient…

In this article we extend a theorem of Andrews, Crippa, and Simon on the asymptotic behavior of polynomials defined by a general class of recursive equations. Here the polynomials are in the variable $q$, and the recursive definition at…