§16.4 Argument Unity

Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.4

§16.4(i) Classification
§16.4(ii) Examples
§16.4(iii) Identities
§16.4(iv) Continued Fractions
§16.4(v) Bilateral Series

§16.4(i) Classification

Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.4.i

The function $\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ is well-poised if

16.4.1 $a_{1}+b_{1}=\dots=a_{q}+b_{q}=a_{{q+1}}+1.$

Symbols:: $q$ : nonnegative integer, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E1
Encodings:: TeX, pMML, png

It is very well-poised if it is well-poised and $a_{1}=b_{1}+1$ .

The special case $\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};1\right)$ is $k$ -balanced if $a_{{q+1}}$ is a nonpositive integer and

16.4.2 $a_{1}+\dots+a_{{q+1}}+k=b_{1}+\dots+b_{q}.$

Symbols:: $q$ : nonnegative integer, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E2
Encodings:: TeX, pMML, png

When $k=1$ the function is said to be balanced or Saalschützian.

§16.4(ii) Examples

Notes:: See Andrews et al. (1999, Chapters 2 and 3) and Slater (1966, §2.3).
Permalink:: http://dlmf.nist.gov/16.4.ii

The function $\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits$ with argument unity and general values of the parameters is discussed in Bühring (1992). Special cases are as follows:

¶ Pfaff–Saalschütz Balanced Sum

Keywords:: generalized hypergeometric functions

16.4.3 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({-n,a,b\atop c,d};1\right)=\frac{\left(c-a\right)_{{n}}\left(c-b\right)_{{n}}}{\left(c\right)_{{n}}\left(c-a-b\right)_{{n}}},$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §18.17(vii)
Permalink:: http://dlmf.nist.gov/16.4.E3
Encodings:: TeX, pMML, png

when $c+d=a+b+1-n$ , $n=0,1,\dots$ . See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity.

¶ Dixon’s Well-Poised Sum

Keywords:: generalized hypergeometric functions

16.4.4 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop a-b+1,a-c+1};1\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}a+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}a-b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b-c+1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E4
Encodings:: TeX, pMML, png

when $\realpart{(a-2b-2c)}>-2$ , or when the series terminates with $a=-n$ :

16.4.5 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({-n,b,c\atop 1-b-n,1-c-n};1\right)=\begin{cases}0,&n=2k+1,\\ \dfrac{(2k)!\mathop{\Gamma\/}\nolimits\!\left(b+k\right)\mathop{\Gamma\/}\nolimits\!\left(c+k\right)\mathop{\Gamma\/}\nolimits\!\left(b+c+2k\right)}{k!\mathop{\Gamma\/}\nolimits\!\left(b+2k\right)\mathop{\Gamma\/}\nolimits\!\left(c+2k\right)\mathop{\Gamma\/}\nolimits\!\left(b+c+k\right)},&n=2k,\\ \end{cases}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $!$ : $n!$ : factorial and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E5
Encodings:: TeX, pMML, png

where $k=0,1,\dots$ .

¶ Watson’s Sum

Keywords:: Watson’s sum, generalized hypergeometric functions

16.4.6 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop\frac{1}{2}(a+b+1),2c};1\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(c+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}(a+b+1)\right)\mathop{\Gamma\/}\nolimits\!\left(c+\frac{1}{2}(1-a-b)\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}(a+1)\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}(b+1)\right)\mathop{\Gamma\/}\nolimits\!\left(c+\frac{1}{2}(1-a)\right)\mathop{\Gamma\/}\nolimits\!\left(c+\frac{1}{2}(1-b)\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E6
Encodings:: TeX, pMML, png

when $\realpart{(2c-a-b)}>-1$ , or when the series terminates with $a=-n$ .

¶ Whipple’s Sum

Keywords:: Whipple’s sum, generalized hypergeometric functions

16.4.7 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,1-a,c\atop d,2c-d+1};1\right)=\frac{\pi\mathop{\Gamma\/}\nolimits\!\left(d\right)\mathop{\Gamma\/}\nolimits\!\left(2c-d+1\right)2^{{1-2c}}}{\mathop{\Gamma\/}\nolimits\!\left(c+\frac{1}{2}(a-d+1)\right)\mathop{\Gamma\/}\nolimits\!\left(c+1-\frac{1}{2}(a+d)\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}(a+d)\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}(d-a+1)\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function and $a,a_{1},\ldots,a_{p}$ : real or complex parameters
Referenced by:: ¶ ‣ §16.4(ii)
Permalink:: http://dlmf.nist.gov/16.4.E7
Encodings:: TeX, pMML, png

when $\realpart{c}>0$ or when $a$ is an integer.

¶ Džrbasjan’s Sum

Keywords:: generalized hypergeometric functions

This is (16.4.7) in the case $c=-n$ :

16.4.8 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({-n,a,1-a\atop d,1-d-2n};1\right)=\frac{\left(\frac{1}{2}(a+d)\right)_{{n}}\left(\frac{1}{2}(d-a+1)\right)_{{n}}}{\left(\frac{1}{2}d\right)_{{n}}\left(\frac{1}{2}(d+1)\right)_{{n}}},$ $n=0,1,\dots$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol and $a,a_{1},\ldots,a_{p}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E8
Encodings:: TeX, pMML, png

¶ Rogers–Dougall Very Well-Poised Sum

Keywords:: generalized hypergeometric functions

16.4.9 $\mathop{{{}_{{5}}F_{{4}}}\/}\nolimits\!\left({a,\frac{1}{2}a+1,b,c,d\atop\frac{1}{2}a,a-b+1,a-c+1,a-d+1};1\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-d+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b-c-d+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b-d+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-c-d+1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E9
Encodings:: TeX, pMML, png

when $\realpart{(b+c+d-a)}<1$ , or when the series terminates with $d=-n$ .

¶ Dougall’s Very Well-Poised Sum

Keywords:: generalized hypergeometric functions

16.4.10 $\mathop{{{}_{{7}}F_{{6}}}\/}\nolimits\!\left({a,\frac{1}{2}a+1,b,c,d,f,-n\atop\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-f+1,a+n+1};1\right)=\frac{\left(a+1\right)_{{n}}\left(a-b-c+1\right)_{{n}}\left(a-b-d+1\right)_{{n}}\left(a-c-d+1\right)_{{n}}}{\left(a-b+1\right)_{{n}}\left(a-c+1\right)_{{n}}\left(a-d+1\right)_{{n}}\left(a-b-c-d+1\right)_{{n}}},$ $n=0,1,\dots$ ,

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E10
Encodings:: TeX, pMML, png

when $2a+1=b+c+d+f-n$ . The last condition is equivalent to the sum of the top parameters plus 2 equals the sum of the bottom parameters, that is, the series is 2-balanced.

§16.4(iii) Identities

Notes:: See Andrews et al. (1999, Chapters 2 and 3) and Slater (1966, §2.4). For (16.4.13) see Donovan et al. (1999).
Keywords:: Racah polynomials, Whipple’s transformation, Wilf–Zeilberger algorithm, generalized hypergeometric functions, $3j,6j,9j$ symbols
Permalink:: http://dlmf.nist.gov/16.4.iii

16.4.11 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};1\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(e\right)\mathop{\Gamma\/}\nolimits\!\left(d+e-a-b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(e-a\right)\mathop{\Gamma\/}\nolimits\!\left(d+e-b-c\right)}\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,d-b,d-c\atop d,d+e-b-c};1\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : real or complex parameters, $b$ : real or complex parameters, $c$ : real or complex parameters, $d$ : real or complex parameters and $e$ : real or complex parameters
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/16.4.E11
Encodings:: TeX, pMML, png

when $\realpart{(d+e-a-b-c)}>0$ and $\realpart{(e-a)}>0$ . The function $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left(a,b,c;d,e;1\right)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. (16.4.11) provides a partial analytic continuation to the region when the only restrictions on the parameters are $\realpart{(e-a)}>0$ , and $d,e$ , and $d+e-b-c\neq 0,-1,\dots$ . A detailed treatment of analytic continuation in (16.4.11) and asymptotic approximations as the variables $a,b,c,d,e$ approach infinity is given by Aomoto (1987).

There are two types of three-term identities for $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ ’s. The first are recurrence relations that extend those for $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ ’s; see §15.5(ii). Examples are (16.3.7) with $z=1$ . Also,

16.4.12 $(a-d)(b-d)(c-d)\left(\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop d+1,e};1\right)-\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};1\right)\right)+abc\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};1\right)=d(d-1)(a+b+c-d-e+1)\left(\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};1\right)-\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop d-1,e};1\right)\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : real or complex parameters, $b$ : real or complex parameters, $c$ : real or complex parameters, $d$ : real or complex parameters and $e$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E12
Encodings:: TeX, pMML, png

and

16.4.13 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};1\right)=\dfrac{c(e-a)}{de}\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b+1,c+1\atop d+1,e+1};1\right)+\dfrac{d-c}{d}\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b+1,c\atop d+1,e};1\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : real or complex parameters, $b$ : real or complex parameters, $c$ : real or complex parameters, $d$ : real or complex parameters and $e$ : real or complex parameters
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/16.4.E13
Encodings:: TeX, pMML, png

Methods of deriving such identities are given by Bailey (1964), Rainville (1960), Raynal (1979), and Wilson (1978). Lists are given by Raynal (1979) and Wilson (1978). See Raynal (1979) for a statement in terms of $3j$ symbols (Chapter 34). Also see Wilf and Zeilberger (1992a, b) for information on the Wilf–Zeilberger algorithm which can be used to find such relations.

The other three-term relations are extensions of Kummer’s relations for $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ ’s given in §15.10(ii). See Bailey (1964, pp. 19–22).

Balanced $\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left(1\right)$ series have transformation formulas and three-term relations. The basic transformation is given by

16.4.14 $\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left({-n,a,b,c\atop d,e,f};1\right)=\frac{\left(e-a\right)_{{n}}\left(f-a\right)_{{n}}}{\left(e\right)_{{n}}\left(f\right)_{{n}}}\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left({-n,a,d-b,d-c\atop d,a-e-n+1,a-f-n+1};1\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $a$ : real or complex parameters, $b$ : real or complex parameters, $c$ : real or complex parameters, $d$ : real or complex parameters and $e$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E14
Encodings:: TeX, pMML, png

when $a+b+c-n+1=d+e+f$ . These series contain $6j$ symbols as special cases when the parameters are integers; compare §34.4.

The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs.

Contiguous balanced series have parameters shifted by an integer but still balanced. One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. See Raynal (1979), Wilson (1978), and Bailey (1964).

A different type of transformation is that of Whipple:

16.4.15 $\mathop{{{}_{{7}}F_{{6}}}\/}\nolimits\!\left({a,\frac{1}{2}a+1,b,c,d,e,f\atop\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-e+1,a-f+1};1\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(a-d+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-e+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-f+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-d-e-f+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-d-e+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-d-f+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-e-f+1\right)}\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left({a-b-c+1,d,e,f\atop a-b+1,a-c+1,d+e+f-a};1\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : real or complex parameters, $b$ : real or complex parameters, $c$ : real or complex parameters, $d$ : real or complex parameters and $e$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E15
Encodings:: TeX, pMML, png

when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. See Bailey (1964, §4.4(4)).

Transformations for both balanced $\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left(1\right)$ and very well-poised $\mathop{{{}_{{7}}F_{{6}}}\/}\nolimits\!\left(1\right)$ are included in Bailey (1964, pp. 56–63). A similar theory is available for very well-poised $\mathop{{{}_{{9}}F_{{8}}}\/}\nolimits\!\left(1\right)$ ’s which are 2-balanced. See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations.

Relations between three solutions of three-term recurrence relations are given by Masson (1991). See also Lewanowicz (1985) (with corrections in Lewanowicz (1987)) for further examples of recurrence relations.

§16.4(iv) Continued Fractions

Keywords:: continued fractions, generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.4.iv

For continued fractions for ratios of $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ functions with argument unity, see Cuyt et al. (2008, pp. 315–317).

§16.4(v) Bilateral Series

Notes:: See Andrews et al. (1999, §2.8) and Slater (1966, Chapter 6).
Keywords:: Dougall’s bilateral sum, bilateral hypergeometric function, bilateral series, generalized hypergeometric functions
Referenced by:: §17.4(ii)
Permalink:: http://dlmf.nist.gov/16.4.v

Denote, formally, the bilateral hypergeometric function

16.4.16 $\mathop{{{}_{{p}}H_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum _{{k=-\infty}}^{\infty}\frac{\left(a_{1}\right)_{{k}}\dots\left(a_{p}\right)_{{k}}}{\left(b_{1}\right)_{{k}}\dots\left(b_{q}\right)_{{k}}}z^{k}.$

Defines:: $\mathop{{{}_{{p}}H_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : bilateral hypergeometric function
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E16
Encodings:: TeX, pMML, png

Then

16.4.17 $\mathop{{{}_{{2}}H_{{2}}}\/}\nolimits\!\left({a,b\atop c,d};1\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(d\right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)\mathop{\Gamma\/}\nolimits\!\left(c+d-a-b-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(d-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)\mathop{\Gamma\/}\nolimits\!\left(d-b\right)},$ $\realpart{(c+d-a-b)}>1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}H_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : bilateral hypergeometric function, $\realpart{}$ : real part, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E17
Encodings:: TeX, pMML, png

This is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).