New integral representations for the Fox-Wright functions and its applications II

Khaled Mehrez

Published 2018-10-20Version 1

Our aim in this paper, is to establish several new integral representations for the Fox--Wright functions ${}_p\Psi_q[^{(\alpha_p,A_p)}_{(\beta_q,B_q)}|z]$ when their terms contain the Fox H-function such that $$\mu=\sum_{j=1}^q\beta_j-\sum_{k=1}^p\alpha_k+\frac{p-q}{2}=-m,\;m\in\mathbb{N}_0.$$ In particular, closed-form integral expressions are derived here for the four parameters Wright function under a special restriction on parameters. Exponential bounding inequalities for a class of the Fox-Wright function ( likes Luke's type inequalities) are derived. Moreover, monotonicity property of ratios involving the Fox-Wright functions are established.