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\begin{equation} \begin{array}{ll} \displaystyle H^\vee_t(u) &\displaystyle = \ \sum_{k=0}^{|u|} \, (-1)^k \, E_{n-k}(u) \, t^{|u|-k} \\ \\ &\text{--- and so ---} \\ \\ \displaystyle F^\vee(z;t) &\displaystyle = \ \sum_{n \geq 0} \, {z^n \over {n!}} \, \langle H^\vee_t \rangle_n \\ &\displaystyle = \ \sum_{n \geq 0} \, {z^n \over {n!}} \, \sum_{k=0}^n \, (-1)^k \, \langle E_k \rangle_n \, t^{n-k} \\ &\displaystyle = \ \sum_{k \geq 0} \, (-z)^k \, \underbrace{\sum_{n \geq 0} \, (zt)^n \, {\langle E_k \rangle_n \over {n!}}}_{\text{$= \, F^\vee_k(zt)$ see below}} \end{array} \end{equation}

\begin{equation} \begin{array}{ll} \displaystyle H^\vee_t(u) &\displaystyle = \ \sum_{k=0}^{|u|} \, (-1)^k \, E_{k}(u) \, t^{|u|-k} \\ \\ &\text{--- and so ---} \\ \\ \displaystyle F^\vee(z;t) &\displaystyle = \ \sum_{n \geq 0} \, {z^n \over {n!}} \, \langle H^\vee_t \rangle_n \\ &\displaystyle = \ \sum_{n \geq 0} \, {z^n \over {n!}} \, \sum_{k=0}^n \, (-1)^k \, \langle E_k \rangle_n \, t^{n-k} \\ &\displaystyle = \ \sum_{k \geq 0} \, (-t)^{-k} \, \underbrace{\sum_{n \geq 0} \, (zt)^n \, {\langle E_k \rangle_n \over {n!}}}_{\text{$= \, F^\vee_k(zt)$ see below}} \end{array} \end{equation}

Question. Does anyone know how to solve the $(\dagger)$-hierarchy of linear recurrences or, equivalently, solve the $(\dagger \dagger)$-hierarchy of 2nd order inhomogeneous ODEs? Have either of these hierarchies been previously studied?

Question. Does anyone know how to solve either the $(\dagger)$-hierarchy of linear recurrences, the $(\dagger \!\dagger)$-hierarchy of 2nd order inhomogeneous ODEs, or equivalently the $(\dagger \! \dagger \! \dagger)$-ODE explained in the second answer/response below? By solve I mean to express the solution in terms of elementary functions, continued fractions, or else by some nice class of special functions (e.g. hypergeometric).