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Concept
Generalized hypergeometric function
Summary
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
A hypergeometric series is formally defined as a power series
in which the ratio of successive coefficients is a rational function of n. That is,
where A(n) and B(n) are polynomials in n.
For example, in the case of the series for the exponential function,
we have:
So this satisfies the definition with A(n) = 1 and B(n) = n + 1.
It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linear factors of the form (aj + n) and (bk + n) respectively, where the aj and bk are complex numbers.
For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality.
The ratio between consecutive coefficients now has the form
where c and d are the leading coefficients of A and B. The series then has the form
or, by scaling z by the appropriate factor and rearranging,
This has the form of an exponential generating function. This series is usually denoted by
or
Using the rising factorial or Pochhammer symbol
this can be written
(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.
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