Generalized Lommel–Wright function and its geometric properties

Abstract The normalization of the combination of generalized Lommel–Wright function $\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)$ J κ 1 , κ 2 κ 3 , m ( z ) ( m ∈ N , $\kappa _{3}>0$ κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by $\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z):=\Gamma ^{m}( \kappa _{1}+1)\Gamma (\kappa _{1}+\kappa _{2}+1)2^{2\kappa _{1}+ \kappa _{2}}z^{1-(\kappa _{2}/2)-\kappa _{1}}\mathcal{J}_{\kappa _{1},\kappa _{2}}^{ \kappa _{3},m}(\sqrt{z})$ J κ 1 , κ 2 κ 3 , m ( z ) : = Γ m ( κ 1 + 1 ) Γ ( κ 1 + κ 2 + 1 ) 2 2 κ 1 + κ 2 z 1 − ( κ 2 / 2 ) − κ 1 J κ 1 , κ 2 κ 3 , m ( z ) , where $\mathcal{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z):=(1-2\kappa _{1}-\kappa _{2})J_{\kappa _{1},\kappa _{2}}^{ \kappa _{3},m}(z)+z ( J_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z) ) ^{\prime}$ J κ 1 , κ