Abstract We generalize the greedy and lazy $\beta $ -transformations for a real base to setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$ , which were recently introduced by first second authors as particular case Cantor bases. As in case, these new transformations, denoted $T_{{\boldsymbol }}}$ $L_{{\boldsymbol respectively, can be iterated order generate digi...

The main aim of this research paper is to introduce a new extension the Gauss hypergeometric function and confluent by using an extended beta function. Some functional relations, summation integral representations, linear transformation formulas, derivative formulas for these functions are derived. We also logarithmic convexity some important inequalities

"The goal of this paper is to study the nonexistence nontrivial solutions following Cauchy problem $$\left\{ \begin{array}{ll} u_{tt}+(-\Delta)^{\beta/2} u+u_{t}=\displaystyle\int_{0}^{t}\left(t-\tau \right) ^{-\gamma}\left\vert u(\tau ,\cdot) \right\vert^{p}d\tau,\\ \cr u(0,x)=u_{0}(x),\quad u_t(0,x)=u_1(x),\quad x\in\mathbb{R}^n, \end{array}\right.$$ where $p>1,\ 0<\gamma <1,\,\, \be...

The dynamics of interacting dark matter-dark energy models is characterized through an interaction rate function quantifying the flow between these sectors. In most functions, expansion Hubble considered and sometimes it argued that, as a local property, inclusion may influence overall dynamics. This starting point present article where we consider very simple cosmic scenario vacuum cold matter...

We introduce the concept of exponentially $s$-convexity in second sense on a time scale interval. prove among other things that if $f: [a, b]\to \mathbb{R}$ is an $s$-convex function, then \begin{align*} &\frac{1}{b-a}\int_a^b f(t)\Delta t\\ &\leq \frac{f(a)}{e_{\beta}(a, x_0) (b-a)^{2s}}(h_2(a, b))^s+\frac{f(b)}{e_{\beta}(b, (b-a)^{2s}}(h_2(b, a))^s, \end{align*} where $\beta$ ...