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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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Rationality of the sum of the reciprocals of the values of a polynomial function at the posi...

A perhaps not so interesting class of examples where the sums in question are known to be either transcendental or explicitly computable algebraics is referenced here: https://mathoverflow.net/a/3358 …
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