The class of  $n$-ary polygroups is a certain subclass of $n$-ary hypergroups, a generalization of D{"o}rnte $n$-arygroups and  a generalization of polygroups. The$beta^*$-relation and the $gamma^*$-relation are the smallestequivalence relations on an $n$-ary polygroup $P$ such that$P/beta^*$ and $P/gamma^*$ are an $n$-ary group and acommutative $n$-ary group, respectively. We use the $beta^*$-...

The best bounds of the form $B(\alpha,\beta,\gamma,x)=(\alpha+\sqrt{\beta^2+\gamma^2 x^2})/x$ for ratios modified Bessel functions are characterized: if $\alpha$, $\beta$ and $\gamma$ chosen in such a way that $B(\alpha,\beta,\gamma,x)$ is sharp approximation $\Phi_{\nu}(x)=I_{\nu-1} (x)/I_{\nu}(x)$ as $x\rightarrow 0^+$ (respectively +\infty$) graphs $\Phi_{\nu}(x)$ tangent at some $x=x_*>0$, ...

Various extensions of classical gamma, beta, Gauss hypergeometric and confluent functions have been proposed recently by many researchers. In this paper, we further generalized extended beta function with some its properties such as symmetric properties, summation formulas, integral representations, connection other special error, Mittag – Leffler, incomplete hypergeometric, Wright, Fox H Meije...

Physics and engineering problems require a detailed knowledge of applied mathematics an understanding special functions such as gamma beta functions. The topic is very important it constantly expanding with the existence new in sciences. In this article, we describe basic theory functions, their connections each other applicability to problems.

In this paper, the new notions of ``belongingness ($in_{gamma}$)"and ``quasi-coincidence ($q_delta$)"  of a fuzzy point with a fuzzyset are  introduced. By means of this new idea, the  concept of$(alpha,beta)$-fuzzy $n$-ary subhypergroup of an $n$-aryhypergroup is given, where $alpha,betain{in_{gamma}, q_{delta},in_{gamma}wedge q_{delta}, ivq}$,  andit is shown that, in 16 kinds of $(alpha,beta...