ways of looking at special functions

Special functions are ways of packing an extraordinarly dense amount of mathematical information in a really tiny thing.  The traditional approach to special functions has emphasized that they arise from solutions to differential equations, and that’s the sort of source where they arise from in many special functions texts. If you treat every special function as the generating function of some combinatorically interesting series, then the massive soggy spaghetti like mess of special functions identities from the point of view of species theory and analytic combinatorics becomes an anvil, so to speak. A special function may satisfy a differential equation and the way that the special function was discovered might have been through the consideration of one differential equation, but the special function may arise in other ways, from the generating function on some combinatorial species, and I think they provide a better window to more interesting functors than by reasoning about combinatorical objects through merely category theoretic ways.

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