A generalized form of the usual Lognormal distribution, denoted with ${}_{\gamma }$, is introduced through the γ-order Normal distribution ${}_{\gamma }$, with its p.d.f. defined into (0,+∞). The study of the c.d.f. of ${}_{\gamma }$ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.

Let $K$ be a finite field of characteristic $p$. Let $K\left(\right(x\left)\right)$ be the field of formal Laurent series $f\left(x\right)$ in $x$ with coefficients in $K$. That is, $$f\left(x\right)=\sum _{n=n_0}^{\infty }{f}_n{x}^n$$ with $n_0\in 𝐙$ and $f_n\in K(n=n_0,n_0+1,\cdots )$. We discuss the distribution of ${\left(\left\{f^m\right\}\right)}_{m=0,1,2,\cdots }$ for $f\in K\left(\right(x\left)\right)$, where $$\left\{f\right\}:=\sum _{n=0}^{\infty }{f}_n{x}^n\in K\left[\left[x\right]\right]$$ denotes the nonnegative part of $f\in K\left(\right(x\left)\right)$. This is a little different from the real number case where the fractional part that excludes constant term (digit of order 0) is considered. We give an alternative proof of a result by De Mathan obtaining the generic...

Let ${}_q$ be a finite field and $A\left(Y\right){\in }_q(X,Y)$. The aim of this paper is to prove that the length of the continued fraction expansion of $A\left(P\right);P{\in }_q\left[X\right]$, is bounded.

The Tracy–Widom $\beta$ distribution is the large dimensional limit of the top eigenvalue of $\beta$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a\to \infty$ the tail of the Tracy–Widom distribution satisfies $$P({\mathrm{𝑇𝑊}}_{\beta }gt;a)=a^{-(3/4)\beta +\mathrm{o}\left(1\right)}exp\left(-\frac{2}{3}\beta a^{3/2}\right).$$

We prove the existence of a limit distribution of the normalized well-distribution measure $W\left(E_N\right)/\sqrt{N}$ (as $N\to \infty$) for random binary sequences $E_N$, by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha =[0;a_1,a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ${\left(a_{\ell }\right)}_{\ell \ge 1}$ of $\alpha$ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

Properties satisfied by the moments of the partial non-central $\chi$-square distribution function, also known as Nuttall Q-functions, and methods for computing these moments are discussed in this paper. The Nuttall Q-function is involved in the study of a variety of problems in different fields, as for example digital communications.

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^m{p}_i\left(n\right)x\left({\tau }_i\left(n\right)\right)=0,n\ge 0\left[\nabla x\left(n\right)-\sum _{i=1}^m{p}_i\left(n\right)x\left({\sigma }_i\left(n\right)\right)=0,n\ge 1\right],$$ where $\left(p_i\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau }_i\left(n\right)$ [${\sigma }_i\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta$ [$\nabla$] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=\tau \left(n\right)}^n{p}_i\left(j\right)>1\left[\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=n}^{\sigma \left(n\right)}{p}_i\left(j\right)>1\right]$$ and $$\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j={\tau }_i\left(n\right)}^{n-1}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\left[\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j=n+1}^{{\sigma }_i\left(n\right)}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\right]$$ are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau }_i\left(n\right)$ $[\sigma \left(n\right)={min}_{1\le i\le m}{\sigma }_i\left(n\right)]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

Let $F\left(X\right)={\sum }_{n\ge 0}{(-1)}^{\epsilon ₙ}{X}^{-\lambda ₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and ${\lambda }_{n+1}/\lambda ₙ>2$. It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that $Q_{\omega }\left(X\right)$ is a polynomial if and only if ω ∈ ℤ. In all the other cases $Q_{\omega }\left(X\right)$ is an infinite formal power series; we discuss...

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form $x={\sum }_{i=1}^{\infty }{c}_i{q}^{-i}$ with integer coefficients $c_i$ satisfying $0\le c_i<q$, i ≥ 1. In this case we say that $\left(c_i\right)=c₁c₂...$ is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also...

Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_G$. We consider the map ${\tau }_{\phi }$ defined on X × G by ${\tau }_{\phi }:(x,g)\mapsto (\tau x,\phi \left(x\right)g)$ and the cocycle ${\left(\phi ₙ\right)}_{n\in \mathbb{Z}}$ generated by φ. Using a characterization of the ergodic invariant measures for ${\tau }_{\phi }$, we give the form of the ergodic decomposition of $\mu \left(dx\right)\otimes m_G\left(dg\right)$ or more generally of the ${\tau }_{\phi }$-invariant measures ${\mu }_{\chi }\left(dx\right)\otimes \chi \left(g\right)m_G\left(dg\right)$, where ${\mu }_{\chi }\left(dx\right)$ is χ∘φ-conformal for an exponential χ on G.