相关论文: Further Investigation on Differential Properties o…
In the paper we consider the Heun functions, which are solutions of the equation introduced by Karl Heun in 1889. The Heun functions generalize many known special functions and appear in many fields of modern physics. Evaluation of the…
This paper introduces and studies a generalization of the $\mathtt{k}$-Bessel function of order $\nu$ given by \[\mathtt{W}^{\mathtt{k}}_{\nu, c}(x):= \sum_{r=0}^\infty \frac{(-c)^r}{\Gamma_{\mathtt{k}}\left( r \mathtt{k}…
We introduce a generalized version of the local Lipschitz number $\textrm{lip}\,u$, and show that it can be used to characterize Sobolev functions $u\in W_{\textrm{loc}}^{1,p}(\mathbb R^n)$, $1\le p\le \infty$, as well as functions of…
Let $\chi$ be an order $c$ multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and $T$-adic Newton polygons of $f$. When $p>(d-e)(2d-1)$, we give a lower bound of…
In this paper, we study the dual of generalized bent functions $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer. It is known…
For a fixed $a \in \{1, 2, 3, \ldots\},$ the radius of starlikeness of positive order is obtained for each of the normalized analytic functions \begin{align*} \mathtt{f}_{a, \nu}(z)&:= \bigg(2^{a \nu-a+1} a^{-\frac{a(a\nu-a+1)}{2}} \Gamma(a…
We introduce the \emph{$\varphi\mathbb{A}$-differentiability} for functions $f:U\subset \mathbb R^{k}\to\mathbb A$ where $\mathbb A$ is the linear space $\mathbb R^{n}$ endowed with an algebra product which is unital, associative,…
A local existence and uniqueness theorem for ODEs in the special algebra of generalized functions is established, as well as versions including parameters and dependence on initial values in the generalized sense. Finally, a Frobenius…
Uniform asymptotic expansions are derived for reverse generalised Bessel polynomials of large degree $n$, real parameter $a$, and complex argument $z$, which are simpler than previously known results. The defining differential equation is…
We start by presenting a generalization of a discrete wave equation that is particularly satisfied by the entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert. The entries are…
Neural ordinary differential equations (neural ODEs) are a popular type of deep learning model that operate with continuous-depth architectures. To assess how well such models perform on unseen data, it is crucial to understand their…
Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number…
It has been an active research issue for many years to construct new bent functions. For $k$ odd with $\gcd(n, k)=1$, and $a\in\mathbb{F}_{3^n}^{*}$, the function $f(x)=Tr(ax^{\frac{3^k+1}{2}})$ is weakly regular bent over…
In this note, we express explicitly the Dunkl kernel and generalized Bessel functions of type $A_{n-1}$ by the Humbert's function $\Phi_{2}^{(n)}$, with one variable specified. The obtained formulas lead to a new proof of Xu's integral…
Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto \delta^{\omega (n)}$, where $\delta \neq 0$ and where $\omega$ counts the number of distinct…
The definite integrals $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^3\D x$, $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^2P_{\nu}(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^3\D x $ and…
The objective of this paper is to establish a general asymptotic representation (\textit{GAR}) for a wide range of statistics, employing two fundamental processes: the functional empirical process (\textit{fep}) and the residual functional…
Let $A = U |A|$ be the polar decomposition of $A$. The Aluthge transform of the operator $A$, denoted by $\tilde{A}$, is defined as $\tilde{A} =|A|^{\frac{1}{2}} U |A|^{\frac{1}{2}}$. In this paper, first we generalize the definition of…