Mellin transform through Euler differential equation and Volterra integral equation

Résumé

We consider the Mellin transform $\mathcal{M}\{f\}(s)$ with $\Re(s)\in [0,1)$. We represent this quantity as the unique initial condition for which a corresponding non-homogeneous linear differential equation with parameter $s$ admits a continuous and bounded solution. We then establish a linear relation between $\mathcal{M}\{f\}(s)$ and its symmetric $\mathcal{M}\{f\}(1-s)$ through an Euler differential equation, and a linear relation between $\mathcal{M}\{f\}(s)$ and its symmetric conjugate $\mathcal{M}\{f\}(1-\overline{s})$ in the form of a Volterra integral equation, whose non-homogeneous term is the same linear combination projected onto the line $\Re(z)=1$. This provides a dynamical reformulation of the Mellin transform in terms of the asymptotic behavior and boundedness of solutions of complex differential and integral equations. In particular, this framework is motivated by the role of Mellin transforms in the analysis of scaling laws and spectral distributions.

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