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 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 an 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. https://doi.org/10.1080/03610927308827073
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. https://doi.org/10.1137/0123026
Janardan, K. 1984. “Moments of Certain Series Distributions and Their Applications.” SIAM Journal on Applied Mathematics
44 (4): 854–68. https://doi.org/10.1137/0144061
Mutafchiev, Ljuben. 1995. “Local Limit Approximations for Lagrangian Distributions.” Aequationes Mathematicae
49 (1): 57–85. https://doi.org/10.1007/BF01827929
Saichev, A., and D. Sornette. 2011. “Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.”
January 28, 2011. http://arxiv.org/abs/1101.5564
Wilf, Herbert S. 1994. Generatingfunctionology
: Academic Press