Recently I have been rereading the Herbert S. Wilf’s free online book Generatingfunctionology – http://www.math.upenn.edu/~wilf/DownldGF.html. It is choke-full of interesting results.
Definitions
For a number series one defines following generating functions (GFs):
- Ordinary Power Series GF (OPSGF): ;
- Exponential Series GF (EGF): ;
- Dirichlet GF (DGF): ;
- Lambert GF (LGF): ;
- Bell GF (BGF): ;
- Poisson GF (PGF): ;
Simple results
- .
- .
- .
- .
- .
- .
- For Fibonacci numbers one has: , leading to ().
- ; which leads to .
- For , $alatex _0=1$ one has: leading to
.
Define as the coefficient next to in power series . Examples and properties:
- ,
- ,
- ,
- ,
- ,
- .
For binomial coeficients:
- ,
- ,
- .
Some orthogonal polynomials:
- Tchebitshev polynomials generating function: .
- Legendre polynomials generating function: .
- Generating function for associated Legendre polynomials: .
Dirichlet series generating functions
- For : .
- For (the Moebius function): .
- For (the zeroth-order divisor function): .
- For (the th-order divisor function): .
- For (the totient function): .
- For (the number of ordered factorizations): .
- For : (the Dirichlet lambda function).
Moebius inversion formula:
- If two DGF series and have coefficient relation , then , and .
- If , then .