The Living Thing / Notebooks :

Generating functions

fancy counting.

Generating functions: A fantastic hack for dealing with counting things, especially, but not only, probabilistic things.

I don’t have much to say here apart from a couple of links and references I need from time to time.

But nor should I; simply download Herbert Wilf’s free and pellucid textbook for all the possible introduction you could need.

Some Rolf Bardeli walks through some beautiful generating function application.

Later I would like to include notes on graph theory and neat count RV based on generating functions. You an rapidly get into weird complex analysis when you consider asymptotics of these guys and get to finally find out why Cauchy’s integral formula and Lagrange’s Inversion Theorem were in your textbook.

Refs

CoSh73
Consul, P. C., & Shenton, L. R.(1973) Some interesting properties of Lagrangian distributions. Communications in Statistics, 2(3), 263–272. DOI.
CoSh72
Consul, P., & Shenton, L. (1972) Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions. SIAM Journal on Applied Mathematics, 23(2), 239–248. DOI.
Jana84
Janardan, K. (1984) Moments of Certain Series Distributions and Their Applications. SIAM Journal on Applied Mathematics, 44(4), 854–868. DOI.
Muta95
Mutafchiev, L. (1995) Local limit approximations for Lagrangian distributions. Aequationes Mathematicae, 49(1), 57–85. DOI.
SaSo11
Saichev, A., & Sornette, D. (2011) Generating functions and stability study of multivariate self-excited epidemic processes. ArXiv:1101.5564 [Cond-Mat, Physics:Physics].
Wilf94
Wilf, H. S.(1994) Generatingfunctionology. . Boston: Academic Press