The hypergeometric series is defined by
where
Of course, the parameters must be such that the denominator factors in the terms of the series are never zero. When one of the numerator parameters equals -n where n is a nonnegative integer this hypergeometric series is a polynomial in z. Otherwise the radius of convergence of the hypergeometric series is given by
A hypergeometric series of the form (0.4.1) is called balanced (or Saalschützian) if r = s + 1, z = 1 and .
The basic hypergeometric series (or q-hypergeometric series) is defined by
where
Again, we assume that the parameters are such that the denominator factors in the terms of the series are never zero. If one of the numerator parameters equals where n is a nonnegative integer this basic hypergeometric series is a polynomial in z. Otherwise the radius of convergence of the basic hypergeometric series is given by
The special case r = s + 1 reads
This basic hypergeometric series was first introduced by Heine in 1846. Therefore it is sometimes called Heine's series. A basic hypergeometric series of this form is called balanced (or Saalschützian) if z = q and .
The q-hypergeometric series is a q-analogue of the hypergeometric series defined by (0.4.1) since
This limit will be used frequently in chapter 5. In all cases the hypergeometric series involved is in fact a polynomial so that convergence is guaranteed.
In the sequel of this paragraph we also assume that each (basic) hypergeometric series is in fact a polynomial. We remark that
In fact, this is the reason for the factors in the definition (0.4.2) of the basic hypergeometric series.
The limit relations between hypergeometric orthogonal polynomials listed in chapter 2 of this report are based on the observations that
and
The limit relations between basic hypergeometric orthogonal polynomials described in chapter 4 of this report are based on the observations that
and
Mostly, the left-hand sides of the formulas (0.4.3) and (0.4.7) occur as limit cases when some numerator parameter and some denominator parameter tend to the same value.
All families of discrete orthogonal polynomials are defined for , where N is a nonnegative integer. In these cases something like (0.4.3) or (0.4.7) occurs in the definition when n = N. In these cases we have to be aware of the fact that we still have a polynomial (in that case of degree N). For instance, if we take n = N in the definition (1.5.1) of the Hahn polynomials we have
and if we take n = N in the definition (3.6.1) of the q-Hahn polynomials we have
So these cases must be understood by continuity.
In cases of discrete orthogonal polynomials we need a special notation for some of the generating functions. We define
for every function f for which , exists. As an example of the use of this Nth partial sum of a power series in t we remark that the generating function (1.10.12) for the Krawtchouk polynomials must be understood as follows : the Nth partial sum of
equals
for .