I don’t have much to say apart from a couple of links and references
I need from time to time.
But nor should I; simply download
Herbert Wilf’s pellucid free textbook
for all the possible
introduction you could need.
Rolf Bardeli walks through some
beautiful generating function application.
Later I would like to include notes on
and neat count RV tricks
based on generating functions.
You can rapidly get into weird complex analysis when you consider
asymptotics of these guys and get to find out why
Cauchy’s integral formula and Lagrange’s Inversion Theorem were in your textbook.
Consul, P. C., and L. R. Shenton. 1973. “Some Interesting Properties of Lagrangian Distributions.” Communications in Statistics
2 (3): 263–72.
Consul, P., and L. Shenton. 1972. “Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions.” SIAM Journal on Applied Mathematics
23 (2): 239–48.
Janardan, K. 1984. “Moments of Certain Series Distributions and Their Applications.” SIAM Journal on Applied Mathematics
44 (4): 854–68.
Mutafchiev, Ljuben. 1995. “Local Limit Approximations for Lagrangian Distributions.” Aequationes Mathematicae
49 (1): 57–85.
Saichev, A., and D. Sornette. 2011. “Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [Cond-Mat, Physics:physics]
Wilf, Herbert S. 1994. Generatingfunctionology
. Boston: Academic Press.