Generalized Hypergeometric Function

Notation

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,

β_n = n!−1 and β_n+1/β_n = 1/(n+1). 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 (a_j + n) and (b_k + n) respectively, where the a_j and b_k 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. The standard notation for this series is

or $\,{}_pF_q \left[\begin{matrix} a_1 & a_2 & \ldots & a_{p} \\ b_1 & b_2 & \ldots & b_q \end{matrix} ; z \right]$

Using the rising factorial or Pochhammer symbol:

this can be written

$\,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\dots(a_p)_n}{(b_1)_n\dots(b_q)_n} \, \frac {z^n} {n!}$

(Note that this use of the Pochhammer symbol is not standard, however it is the standard usage in this context.)

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Generalized Hypergeometric Function - Notation