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Definitions

Hypergeometric identity:

$\displaystyle \sum_{k=-\infty}^{\infty} F(n,k) = \mathrm{rhs}(n),$

where both $\frac{F(n+1,k)}{F(n,k)}$ and $\frac{F(n,k+1)}{F(n,k)}$ are rational functions of

and

Standard form for hypergeometic series:

$\displaystyle {}_{r}\mathrm{F}_{s} \left[ \genfrac{}{}{0pt}{}{ a_1, a_2, \dots,... ...ac { (a_1)_k (a_2)_k \cdots (a_r)_k } { (b_1)_k (b_2)_k \cdots (b_s)_k k!} z^k,$

(1.1)

where

$\displaystyle (a)_k \overset{\mathrm{def}}{=} a(a+1)(a+2)\cdots(a+k-1);$
$\displaystyle (a)_0 = 1,$

can be written as

From (1.1), the ratio

$\displaystyle \frac{F(n,k+1)}{F(n,k)} = \frac{(k+a_1)(k+a_2)\cdots(k+a_r)}{(k+b_1)(k+b_2)\cdots (k+b_s)(k+1)}z.$

Math Guest 1 2004-08-10