An Application of Pascal Distribution Series on Certain Analytic Functions Associated with Stirling Numbers and Sălăgean Operator

Abstract

In the present paper, we will observe that the Sălăgean differential operator can be written in terms of Stirling numbers. Furthermore, we find a necessary and sufficient condition and inclusion relation for Pascal distribution series to be in the class ${\mathrm{\mathbb{P}}}_k\left(\lambda ,\alpha \right)$ of analytic functions with negative coefficients defined by the Sălăgean differential operator. Also, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.