Integral transforms and ordinary differential equations of infinite order

We review our work on ordinary differential equations of infinite order and we apply our theory to equations defined with the help of the Riemann zeta function which are of interest for modern theoretical physics. We interpret these equations with the help of the Laplace and Borel transforms, we study existence, uniqueness and regularity of solutions to the equation f ( ∂ t ) ϕ = J ( t ) , t ≥ 0 , $$ \begin{aligned} f(\partial _t) \phi = J(t) \; , \; \; \; t \ge 0, \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/um36.tif"/>

and we analyze the delicate issue of the initial value problem.