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I'm familiar with the concept of Exponential Family as it appears in probability theory (see e.g. the wikipedia page). Lately, while reading "generatingfunctionology" by H. Wilf, I stumbled into something which goes by the same name Exponential Family (definition is at page 75) and has to do with counting stuff, for instance the number of labeled, connected graphs with $n$ vertices.

Question: how are the two concepts related? I'm assuming thy are, since they share the same name, but after some thought and internet reaserch I couldn't figure it out.

A guess: could the idea of generating function be the fil rouge? Normalization constants of exponential families naturally lend themselves to be turned into generating functions for moments and cumulants.

long_john
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I think they are unrelated. The first definition (parametric set of probability distributions) is far more common.

I feel like Wilf's usage of this term in that textbook is not widely adopted in the context of EGFs. For instance, the other main reference on generating functions is Flajolet/Sedgewick [PDF] who does not use "exponential family" at all when discussing EGFs. (In fact, the only usage of that term in this latter text is using the other "collection of probability distributions" definition.)

angryavian
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