We study the triangular array defined by Graham–Knuth–Patashnik recurrence $T(n,k) = (\alpha n + \beta k \gamma)\, T(n-1,k)+(\alpha' \beta' \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) \delta_{k0}$ and parameters $\mathbf{\mu} (\alpha,\beta,\gamma, \alpha',\beta',\gamma')$. show that family of arrays $T(\mathbf{\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \...

Abstract In this article, we study the logarithmic Mahler measure of one-parameter family $$Q_\alpha=y^2+(x^2-\alpha x)y+x,$$ denoted by $\mathrm{m}(Q_\alpha)$. The zero loci Qα generically define elliptic curves Eα, which are 3-isogenous to Hessian curves. We particularly interested in case $\alpha\in (-1,3)$, has not been considered literature due certain subtleties. For α interval, establish...

The present work constitutes a first step towards establishing systematic framework for treating variational problems that depend on given input function through mixture of its derivatives different orders in directions. For fixed vector \(\mathbf {a} := (a_1, \ldots , a_N) \in \mathbb {N}^N\) and \(u :\mathbb {R}^N \supset \varOmega \rightarrow {R}^n\) we denote by \(\nabla _{\mathbf {a}} u (\...

We study the position distribution $P(\vec{R},N)$ of a run-and-tumble particle (RTP) in arbitrary dimension $d$, after $N$ runs. assume that constant speed $v>0$ during each running phase is independently drawn from probability $W(v)$ and direction chosen isotropically tumbling. The clearly isotropic, $P(\vec{R},N)\to P(R,N)$ where $R=|\vec{R}|$. show that, under certain conditions on $d$ for l...

The minimal excludant, or “mex” function, on a set S of positive integers is the least integer not in S. In recent paper, Andrews and Newman extended mex-function to partitions found numerous surprising partition identities connected with these functions. Very recently, da Silva Sellers present parity considerations one families functions studied, namely $$p_{t,t}(n)$$ , provide complete charac...

Let $${\mathbb {F}}_{q^n}$$ be a finite field with $$q^n$$ elements and r positive divisor of $$q^n-1$$ . An element $$\alpha \in {\mathbb {F}}_{q^n}^*$$ is called r-primitive if its multiplicative order $$(q^n-1)/r$$ Also, k-normal over {F}}_q$$ the greatest common polynomials $$g_{\alpha }(x) = \alpha x^{n-1}+ ^q x^{n-2} + \ldots ^{q^{n-2}}x ^{q^{n-1}}$$ $$x^n-1$$ in {F}}_{q^n}[x]$$ has degre...

A Piatetski-Shapiro sequence with exponent $\alpha$ is a of integer parts $n^\alpha$ $(n = 1,2,\ldots)$ non-integral $\alpha > 0$. We let $\mathrm{PS}(\alpha)$ denote the set those terms. In this article, we study so that equation $ax + by cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(\alpha)^3$, and give lower bound for its Hausdorff dimension. As corollary, find...

We prove a new result about the mutual behaviour of irrationality measure functions $$\psi _{\alpha _j}(t)$$ for n different real numbers $$\alpha _j, j=1,...,n$$ .