Generalized Hypergeometric Function

The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written

(The factor of 👁 k+1
in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written

where 👁 (a)_k
is the Pochhammer symbol or rising factorial

A generalized hypergeometric function 👁 _pF_q(a_1,...,a_p;
👁 b_1,...,b_q;x)
therefore has 👁 p
parameters of type 1 and 👁 q
parameters of type 2.

A number of generalized hypergeometric functions has special names. 👁 _0F_1(;b;z)
is called a confluent hypergeometric limit function, and is implemented in the Wolfram Language as [b, z]. 👁 _1F_1(a;b;z)
(also denoted 👁 M(z)
) is called a confluent hypergeometric function of the first kind, and is implemented in the Wolfram Language as [a, b, z]. The function 👁 _2F_1(a,b;c;z)
is often called "the" hypergeometric function or Gauss's hypergeometric function, and is implemented in the Wolfram Language as [a, b, c, x]. Arbitrary generalized hypergeometric functions are implemented as [👁 {
a1, ...ap👁 }
, 👁 {
b1, ..., bq👁 }
, x].

The notation for generalized hypergeometric functions was introduced by Pochhammer in 1870 and modified by Barnes (1907, 1908ab; Slater 1960, p. 2; Hardy 1999, p. 111). A number of notational variations are commonly used, including

used primarily for 👁 p,q<=2
, using square brackets instead of parentheses

(Slater 1960, p. 2), including 👁 x
at the end of the first row and aligning slots in the second row from the right

(Bailey 1935, p. 9), including 👁 x
at the end of the first row and centering each row

(Bailey 1935, p. 14), using strict matrix-like alignment of each column with columns right-aligned along their right-most elements

or

(Slater 1960, p. 31), and a variation in which rows are centered and 👁 x
is placed to the right separated by a vertical bar and using parenthesis

(Graham et al. 1994, p. 205) or using square brackets and a semicolon

The latter convention will be used in this work, as it provides the clearest delineation of the argument 👁 x
while making the most sparing use of white space in the typesetting of expressions that may contain a large number of symbolic parameters of differing lengths.

If the argument is equal to 👁 x=1
, then it is conventional to omit the argument altogether, although the trailing semicolon may be either retained or also discarded depending on notational convention. Bailey (1935, p. 9) uses the notation

although in this work, the semicolon will be omitted, i.e.,

The Kampe de Feriet function is a generalization of the generalized hypergeometric function to two variables.

The generalized hypergeometric function 👁 F_n(x)=_pF_q[a_1,a_2...,a_p; b_1,b_2,...,b_q;x]
satisfies

for any of its numerator parameters 👁 n=alpha_k
, and

for any of its denominator parameters 👁 n=beta_k
, where

is the differential operator (Rainville 1971, Koepf 1998, p. 27).

The generalized hypergeometric function

is a solution to the differential equation

(Bailey 1935, p. 8). The other linearly independent solution is

A generalized hypergeometric function 👁 _(q+1)F_q
converges absolutely on the unit circle if

(Rainville 1971, Koepf 1998).

Many sums can be written as generalized hypergeometric functions by inspection of the ratios of consecutive terms in the generating hypergeometric series. For example, for

the ratio of successive terms is

yielding

(Petkovšek et al. 1996, pp. 44-45).

Gosper (1978) discovered a slew of unusual hypergeometric function identities, many of which were subsequently proven by Gessel and Stanton (1982). An important generalization of Gosper's technique, called Zeilberger's algorithm, in turn led to the powerful machinery of the Wilf-Zeilberger pair (Zeilberger 1990).

Special hypergeometric identities include Gauss's hypergeometric theorem

for 👁 R[c-a-b]>0
, Kummer's formula

where 👁 a-b+c=1
and 👁 b
is a positive integer, Saalschütz's theorem

for 👁 d+e=a+b+c+1
with 👁 c
a negative integer and 👁 (a)_n
the Pochhammer symbol, Dixon's theorem

where 👁 1+a/2-b-c
has a positive real part, 👁 d=a-b+1
, and 👁 e=a-c+1
, the Clausen formula

for 👁 a+b+c-d=1/2
, 👁 e=a+b+1/2
, 👁 a+f=d+1=b+g
, 👁 d
a nonpositive integer, and the Dougall-Ramanujan identity

where 👁 n=2a_1+1=a_2+a_3+a_4+a_5
, 👁 a_6=1+a_1/2
, 👁 a_7=-n
, and 👁 b_i=1+a_1-a_(i+1)
for 👁 i=1
, 2, ..., 6. For all these identities, 👁 (a)_n
is the Pochhammer symbol.

Gessel (1995) found a slew of new identities using Wilf-Zeilberger pairs, including the following:

(Petkovšek et al. 1996, pp. 135-137).

The following table gives various named identities ordered by the orders 👁 (p,q)
of the 👁 _pF_q
s they involve. Bailey (1935) gives a large number of such identities.

Nørlund (1955) gave the general transformation

where 👁 (a)_n
is the Pochhammer symbol. This identity is based on the transformation due to Euler

where 👁 Delta
is the forward difference and

(Nørlund 1955).

See also

Carlson's Theorem, Clausen Formula, Confluent Hypergeometric Function of the First Kind, Confluent Hypergeometric Limit Function, Dixon's Theorem, Dougall-Ramanujan Identity, Dougall's Theorem, Generalized Hypergeometric Differential Equation, Gosper's Algorithm, Hypergeometric Function, Hypergeometric Identity, Hypergeometric Series, Jackson's Identity, k-Balanced, Kampe de Feriet Function, Kummer's Theorem, Lauricella Functions, Nearly-Poised, q-Hypergeometric Function, Ramanujan's Hypergeometric Identity, Saalschütz's Theorem, Saalschützian, Sister Celine's Method, Slater's Formula, Thomae's Theorem, Watson's Theorem, Well-Poised, Whipple's Identity, Whipple's Transformation, Wilf-Zeilberger Pair, Zeilberger's Algorithm

Related Wolfram sites

http://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/, http://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQRegularized/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F0/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F2/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F3/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric3F2/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric4F3/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric5F4/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric6F5/

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References

Bailey, W. N. "Some Identities Involving Generalized Hypergeometric Series." Proc. London Math. Soc. Ser. 2 29, 503-516, 1929.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.Barnes, E. W. "The Asymptotic Expansion of Integral Functions Defined by Generalised Hypergeometric Series." Proc. London Math. Soc. 5, 59-116. 1907.Barnes, E. W. "On Functions Defined by Simple Hypergeometric Series." Trans. Cambridge Philos. Soc. 20, 253-279, 1908a.Barnes, E. W. "A New Development of the Theory of Hypergeometric Functions." Proc. London Math. Soc. 6, 141-177, 1908b.Dwork, B. Generalized Hypergeometric Functions. Oxford, England: Clarendon Press, 1990.Exton, H. Multiple Hypergeometric Functions and Applications. New York: Wiley, 1976.Exton, H. Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, 1978.Gessel, I. "Finding Identities with the WZ Method." J. Symb. Comput. 20, 537-566, 1995.Gessel, I. and Stanton, D. "Strange Evaluations of Hypergeometric Series." SIAM J. Math. Anal. 13, 295-308, 1982.Gosper, R. W. "Decision Procedures for Indefinite Hypergeometric Summation." Proc. Nat. Acad. Sci. USA 75, 40-42, 1978.Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Hypergeometric Functions." §5.5 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 204-216 1994.Hardy, G. H. "Hypergeometric Series." Ch. 7 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 101-112, 1999.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/.Klein, F. Vorlesungen über die hypergeometrische Funktion. Berlin: J. Springer, 1933.Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its 👁 q
-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, 1-168, 1998.Koepf, W. "Hypergeometric Database." Ch. 3 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 12 and 31-43, 1998.Nørlund, N. E. "Hypergeometric Functions." Acta Math. 94, 289-349, 1955.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Rainville, E. D. Special Functions. New York: Chelsea, 1971.Saxena, R. K. and Mathai, A. M. Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. New York: Springer-Verlag, 1973.Slater, L. J. Confluent Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1960.Slater, L. J. Generalized Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1966.Zeilberger, D. "A Fast Algorithm for Proving Terminating Hypergeometric Series Identities." Discrete Math. 80, 207-211, 1990.

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Generalized Hypergeometric Function

Cite this as:

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

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