Related papers: Generating functions for the powers in $\text{GL}(…

The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse…

In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The…

Let $q$ be an odd prime and $k$ be a natural number. We show that a finite subset of integers $S$ that does not contain any perfect $q^{th}$ power, contains a $q^{th}$ power residue modulo almost every natural numbers $N$ with at most $k$…

We find the normal form of nilpotent elements in semisimple Lie algebras that generalizes the Jordan normal form in $\mathfrak{sl}_N$, using the theory of cyclic elements.

In this paper, we give a simple criterion to verify that functions of the form $e^g$ are in the Hayman class when $g$ is a power series with nonnegative coefficients. Thus, using the Hayman and B\'aez-Duarte formulas, we obtain asymptotics…

Let m be a fixed positive integer. It is well-known that a permutation $\sigma$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of \sigma. Let p_m(n)…

In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in R^m. Here spherical monogenics are polynomial solutions of the Dirac equation in R^m. In particular, we obtain…

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…

In this paper, we construct a binary linear code connected with the Kloosterman sum for $GL(2,q)$. Here $q$ is a power of two. Then we obtain a recursive formula generating the power moments 2-dimensional Kloosterman sum, equivalently that…

We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that…

The first known families of cages arised from the incidence graphs of generalized polygons of order $q$, $q$ a prime power. In particular, $(q+1,6)$--cages have been obtained from the projective planes of order $q$. Morever, infinite…

It was recently shown that $q\omega(q)$, where $\omega(q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less…

We study three different $q$-analogues of the harmonic numbers. As applications, we present some generating functions involving number theoretical functions and give the $q$-generalization of Gosper's exponential generating function of…

For any positive integers $n=2k$ and $m$ such that $m\geq k$, in this paper we show the maximal number of bent components of any $(n,m)$-functions is equal to $2^{m}-2^{m-k}$, and for those attaining the equality, their algebraic degree is…

We construct a $k$-fold $q$-series as a generating function of $k$-regular partitions for each positive integer $k$. The $k=1$ case is one of Euler's $q$-series identities pertaining to the partitions into distinct parts. The construction…

Word maps have been studied for matrix groups over a field. We initiate the study of problems related to word maps in the context of the group $\mathrm{GL}_n(\mathscr O_2)$, where $\mathscr O_2$ is a finite local principal ideal ring of…

We study the number of generators of ideals in regular rings and ask the question whether $\mu(I)<\mu(I^2)$ if $I$ is not a principal ideal, where $\mu(J)$ denotes the number of generators of an ideal $J$. We provide lower bounds for the…

We consider the relations of generalized commutativity in the algebra of formal series $ M_q (x^i ) $, which conserve a tensor $ I_q $-grading and depend on parameters $ q(i,k) $ . We choose the $ I_q $-preserving version of differential…

Let $\Gamma \,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer, $5^{th}$ power-free. Let $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $\zeta_5$, and…

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs…