Hypergeometric Series

A hypergeometric series $\sum_k c_k$ is a series for which $c_0=1$ and the ratio of consecutive terms is a Rational Function of the summation index $k$, i.e., one for which

$${c_{k+1}\over c_k}={P(k)\over Q(k)},$$ (1)

with $P(k)$ and $Q(k)$ Polynomials. The functions generated by hypergeometric series are called Hypergeometric Functions or, more generally, Generalized Hypergeometric Functions. If the polynomials are completely factored, the ratio of successive terms can be written

$${c_{k+1}\over c_k}={P(k)\over Q(k)}={(k+a_1)(k+a_2)\cdots(k+a_p)\over (k+b_1)(k+b_2)\cdots(k+b_q)(k+1)}x,$$ (2)

where the factor of $k+1$ in the Denominator is present for historical reasons of notation, and the resulting Generalized Hypergeometric Function is written

$${}_pF_q\left[{\matrix{a_1 & a_2 & \cdots & a_p\cr b_1 & b_2 & \cdots & b_q\cr}; x}\right]=\sum_{k=0} c_k x^k.$$ (3)

If $p=2$ and $q=1$, the function becomes a traditional Hypergeometric Function ${}_2F_1(a,b;c;x)$.

Many sums can be written as Generalized Hypergeometric Functions by inspections of the ratios of consecutive terms in the generating hypergeometric series.

See also Generalized Hypergeometric Function, Geometric Series, Hypergeometric Function, Hypergeometric Identity

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. ``Hypergeometric Series,'' ``How to Identify a Series as Hypergeometric,'' and ``Software That Identifies Hypergeometric Series.'' §3.2-3.4 in A=B. Wellesley, MA: A. K. Peters, pp. 34-42, 1996.