The Parabolic Mellin Transform: Gamma and Zeta Integral Representations

Peter Reinhard Hansen, Chen Tong

We introduce the Parabolic Mellin Transform (PMT), defined by $P_{\ddot{I}}ƒ[f](z)={\int }_{-\infty }^{\infty }{w}^{2z}f(w^2)dt$, where $w=\ddot{I}ƒ+it$ and $\ddot{I}ƒ>0$. Under the substitution $u=w^2$, the…