Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSums of polynomial-type exceptional units modulo $n$
96
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $f(x)inmathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $nge 1$ and $kge 2$. An integer $a$ is called an $f$-exunit in the ring $mathbb{Z}_n$ of residue classes modulo $n$ if $gcd(f(a),n)=1$. In this paper, we use the principle of cross-classification to derive an explicit formula for the number ${mathcal N}_{k,f,c}(n)$ of solutions $(x_1,...,x_k)$ of the congruence $x_1+...+x_kequiv cpmod n$ with all $x_i$ being $f$-exunits in the ring $mathbb{Z}_n$. This extends a recent result of Anand {it et al.} [On a question of $f$-exunits in $mathbb{Z}/{nmathbb{Z}}$, {it Arch. Math. (Basel)} {bf 116} (2021), 403-409]. We derive a more explicit formula for ${mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.
rate research
Read More
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvys paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2ldots$. We estimate the orders of these roots of unity in terms of the degrees and the heights of the polynomials $c_i$ and $f_i$.
Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were earlier conjectured by the same authors and involve $q$-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved $q$-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.
In this paper, we study the alternating Euler $T$-sums and $S$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $S$-sums, from which, the parity theorems of Hoffmans double and triple $t$-values and Kaneko-Tsumuras double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $S$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.
By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Phi_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
