Kravchuk polynomials

Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname "Кравчу́к") are discrete orthogonal polynomials associated with the binomial distribution, introduced by .
The first few polynomials are (for "q"=2):
The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
Definition.
For any prime power "q" and positive integer "n", define the Kravchuk polynomial
Properties.
The Kravchuk polynomial has the following alternative expressions:
Symmetry relations.
For integers formula_8, we have that 
Orthogonality relations.
For non-negative integers "r", "s",
Generating function.
The generating series of Kravchuk polynomials is given as below. Here formula_11 is a formal variable.