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

with 👁 P(k)
and 👁 Q(k)
polynomials. In this case, 👁 c_k
is called a hypergeometric term (Koepf 1998, p. 12). 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

where the factor of 👁 k+1
in the denominator is present for historical reasons of notation, and the resulting generalized hypergeometric function is written

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

Binomial Sums, Generalized Hypergeometric Function, Geometric Series, Hypergeometric Function, Hypergeometric Identity, Hypergeometric Term

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References

Ishkhanyan, T. "Hypergeometric Functions: From Euler to Appell and Beyond." Jan. 25, 2024. https://blog.wolfram.com/2024/01/25/hypergeometric-functions-from-euler-to-appell-and-beyond/.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Petkovšek, 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. http://www.cis.upenn.edu/~wilf/AeqB.html.

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Hypergeometric Series

Cite this as:

Weisstein, Eric W. "Hypergeometric Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HypergeometricSeries.html

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