In this paper, we develop theorems on finite and infinite summation formulas by utilizing the q (q,h) anti-difference operators, also extend these core to q(α) (q,h)α difference operators. Several integer order based operator have been published, which gave us idea derive fractional equations for construct a function known as quantum geometric alpha-quantum function, behaves class of series. We...

Our goal in this article is to use ideas from symmetric q-calculus operator theory the study of meromorphic functions on punctured unit disc and propose a novel q-difference for these functions. A few additional classes are then defined light new operator. We prove many useful conclusions regarding newly constructed functions, such as convolution, subordination features, integral representation...

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $ell^1(S)$ is  a symmetric $ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigr...