In this article, the multi-parameters Mittag-Leffler function is studied in detail. As a consequence, a series of novel results such as the integral representation, series representation and Mellin transform to the above function, are obtained. Especially, we associate the multi-parameters Mittag- Leffler function with two special functions which are the generalized Wright hypergeometric and the Fox's-H functions. Meanwhile, some interesting integral operators and derivative operators of this function, are also discussed.

This paper is devoted to the study of Tur\’{an} type inequalities for some well-known special functions such as supersine and supercosine which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality.

The element model of the viscoelastic constitutive model has its significant advantage in describing the three stages of rheological curve. Through the medium of the k-Hilfer-Prabhakar fractional derivative, our primary intension in this paper is to establish a viscoelastic constitutive model coupling with the nonlinear time varying elastic element E(t). The k-Hilfer-Prabhakar fractional derivative is characterized by the recovery between the modified model and the known classical models of linear viscoelasticity. According to the theory of the Laplace transform and Boltzmann superposition principle, the strain of the proposed model is obtained. In summing up it may be stated that the analysis of parameter identification indicates the validity and rationality of the modified model.