Related papers: Functional identities of one variable

We show that there exists $k \in \bbn$ and $0 < \e \in\bbr$ such that for every field $F$ of characteristic zero and for every $n \in \bbn$, there exists explicitly given linear transformations $T_1,..., T_k: F^n \to F^n$ satisfying the…

Let $p$ be an odd prime and $\F_q$ be the finite field with $q=p^n$ elements. A planar function $f:\F_q\rightarrow\F_q$ is called homogenous if $f(\lambda x)=\lambda^df(x)$ for all $\lambda\in\F_p$ and $x\in\F_q$, where $d$ is some fixed…

We study algebras $A,$ over a field of characteristic zero, satisfying $(x^p, x^q, x^r)=0$ for $p, q, r$ in ${1, 2}.$ The existence of a unit element in such algebras leads to the third power-associativity. If, in addition, $A$ has degree…

We explore a number of functional properties of the $q$-gamma function and a class of its quotients; including the $q$-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions…

Let F be a finite extension of Q_p. We show that every Schwartz function on F, with values in an algebraic closure of Q_p, is the uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof…

We consider the algebra $\square_q$ which is a mild generalization of the quantum algebra $U_q(\frak{sl}_2)$. The algebra $\square_q$ is defined by generators and relations. The generators are $\{x_i\}_{i\in \mathbb{Z}_4}$, where…

It is shown that if $A$ is a uniform algebra generated by real-analytic functions on a suitable compact subset $K$ of a real-analytic variety such that the maximal ideal space of $A$ is $K$, and every continuous function on $K$ is locally a…

Given any separable complex Hilbert space, any trace-class operator $B$ which does not have purely imaginary trace, and any generator $L$ of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a…

This paper is motivated by an open question in $p$-adic Fourier theory, that seems to be more difficult than it appears at first glance. Let $L$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $o_L$ and let $\mathbb{C}_p$…

We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose $q$-expansions satisfy \[ f_k(A, \tau) \colon = q^{-k}(1+a(1)q+a(2)q^2+...) + O(q),\] where $a(n)$ are…

In this paper, we tackle unresolved inquiries by Ferreira et al. \cite{bruno} in their recent publication, ``Functional Identity on Division Algebras". We delve into the intricate behavior of additive functions on matrix algebras over…

Zagier's well-known work on traces of singular moduli relates the coefficients of certain weakly holomorphic modular forms of weight $1/2$ to traces of values of the modular $j$-function at imaginary quadratic points. A real quadratic…

Let $\overline{\mathbb Q}$ be an algebraic closure of $\mathbb Q$ and let $A$ denote the ring of algebraic integers in $\overline{\mathbb Q}$. If $\mathcal S = \overline{\mathbb Q}^\times/A^\times$ then $\mathcal S$ is a vector space over…

Let $\mathbb F$ denote an algebraically closed field and assume that $q\in \mathbb F$ is a primitive $d^{\rm \, th}$ root of unity with $d\not=1,2,4$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $n\geq2$ a positive integer, $\mathbb{V}_0$ a $n$-dimensional vector space over $\mathbb{F}_q$ and $\mathbb{T}_0$ the set of all linear functionals from $\mathbb{V}_0$ to…

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass…

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications.…

Let \begin{equation*} A_{q}^{(\alpha)}(a;z) = \sum_{k=0}^{\infty} \frac{(a;q)_{k} q^{\alpha k^2} z^k} {(q;q)_{k}}, \end{equation*} where $\alpha >0,~0<q<1.$ In a paper of Ruiming Zhang, he asked under what conditions the zeros of the entire…

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…

It is known that, under certain conditions, *-representability and extensibility to the unitized *-algebra of a positive linear functional, defined on a *-algebra without unit, are equivalent. In this paper, a new condition for an analogous…