Fine Tuning Points of Generating Function Construction in Integral Form for Linear Recursions

In our a previous work we have dealt with the generating function construction for representing the general term of a sequence as a moment like integral where a generating function takes the role of a weight function. We have assumed therein that the each pair of sequence elements satisfy a first order homogeneous linear recursion with variant coefficients. Then we have tried to construct ODE (s) whose solutions under appropriate boundary conditions give the generating function uniquely if the recursion is accompanied by an appropriate initial condition. We could have been able to achieve the goal and discussed the solutions and their behaviors in certain level of details. This work aims almost the same thing as the previous work but this time not via an ODE, instead, an integral equation.