On generalized Bessel–Maitland function

Abstract An approach to the generalized Bessel–Maitland function is proposed in present paper. It denoted by $\mathcal{J}_{\nu , \lambda }^{\mu }$ Jν,λμ where $\mu >0$ xmlns:mml="http://www.w3.org/1998/Math/MathML">μ>0 and $\lambda ,\nu \in \mathbb{C\ xmlns:mml="http://www.w3.org/1998/Math/MathML">λ,ν∈C get increasing interest from both theoretical mathematicians applied scientists. The main objective establish integral representation of ,\lambda applying Gauss’s multiplication theorem for beta as well Mellin–Barnes using residue theorem. Moreover, m th derivative considered, it turns out that expressed Fox–Wright function. In addition, recurrence formulae other identities involving derivatives are derived. Finally, monotonicity ratio between two modified functions $\mathcal{I}_{\nu xmlns:mml="http://www.w3.org/1998/Math/MathML">Iν,λμ defined }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu }(iz)$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Iν,λμ(z)=i−2λ−νJν,λμ(iz) a different order, hyperbolic functions, some results }(z)$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Iν,λμ(z) obtained idea proofs comes quotient Maclaurin series. As an application, inequalities (like Turán-type their reverse) proved. Further investigations on this underway will be reported forthcoming