stylistic:tone, linguistic:paraphrasing
Hypergeometric-Gaussian modes

u p m (ρ, ϕ, Z) = 2 p + | m | + 1 π Γ (p + | m | + 1) Γ (p 2 + | m | + 1) Γ (| m | + 1) i | m | + 1 × Z p 2 (Z + i) − (p 2 + | m | + 1) ρ | m | × exp (− i ρ 2 Z + i) e i m ϕ 1 F 1 (− p 2, | m | + 1; ρ 2 Z (Z + i)) {\displaystyle {\begin{aligned}u_{{\mathsf {p}}m}(\rho,\phi,\mathrm {Z}){}={}&{\sqrt {\frac {2^{{\mathsf {p}}+|m|+1}}{\pi \Gamma ({\mathsf {p}}+|m|+1)}}}\;{\frac {\Gamma \left({\frac {\mathsf {p}}{2}}+|m|+1\right)}{\Gamma (|m|+1)}}\,i^{|m|+1}\times {}\\&\mathrm {Z} ^{\frac {\mathsf {p}}{2}}\,(\mathrm {Z} +i)^{-\left({\frac {\mathsf {p}}{2}}+|m|+1\right)}\,\rho ^{|m|}\times {}\\&\exp \left(-{\frac {i\rho ^{2}}{\mathrm {Z} +i}}\right)\,e^{im\phi }\,{}_{1}F_{1}\left(-{\frac {\mathsf {p}}{2}},|m|+1;{\frac {\rho ^{2}}{\mathrm {Z} (\mathrm {Z} +i)}}\right)\end{aligned}}}