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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
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Solution to difference differential equation with constant coefficients
Thanks to Michael Renardy for the analogy with the heat equation. Let's assume $q_i = q$, i.e. independent on $i$. The difference-differential equation can then be solved by using generating function …
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Solution to difference differential equation with constant coefficients
This problem arose when solving a continuous Markov chain exercise from a book I am studying. Given a set of positive $q_i$ with $i \in \mathbb{Z} $, and non-negative $\lambda$ and $\mu$ that add to 1 …