# Dougall bilateral sum

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##### 1: 16.4 Argument Unity
###### Dougall’s Very Well-Poised Sum
Denote, formally, the bilateral hypergeometric function …This is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).
##### 2: 15.4 Special Cases
###### Dougall’s BilateralSum
15.4.25 $\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{% \Gamma\left(c+n\right)\Gamma\left(d+n\right)}=\frac{\pi^{2}}{\sin\left(\pi a% \right)\sin\left(\pi b\right)}\*\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma% \left(c-a\right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b% \right)}.$
##### 3: 17.1 Special Notation
###### §17.1 Special Notation
The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, and the $q$-analogs of the Appell functions $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)$, $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)$, $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)$, and $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)$. …
$f(\chi_{1};\chi_{2},\dots,\chi_{n})+\operatorname{idem}\left(\chi_{1};\chi_{2}% ,\dots,\chi_{n}\right)=\sum_{j=1}^{n}f(\chi_{j};\chi_{1},\chi_{2},\dots,\chi_{% j-1},\chi_{j+1},\dots,\chi_{n}).$
##### 4: 14.18 Sums
###### Dougall’s Expansion
For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …
##### 5: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
###### Sum Related to (17.6.4)
For similar formulas see Verma and Jain (1983).
##### 8: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions
###### §17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions
17.10.1 ${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),$
17.10.2 ${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).$
17.10.3 ${{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),$
17.10.4 ${{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.$
##### 9: 17.4 Basic Hypergeometric Functions
###### §17.4 Basic Hypergeometric Functions
In these references the factor $\left((-1)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}\right)^{s-r}$ is not included in the sum. …
###### §17.4(ii) ${{}_{r}\psi_{s}}$ Functions
17.4.3 ${{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.$
17.4.4 $\lim_{q\to 1-}{{}_{r}\psi_{r}}\left({q^{a_{1}},q^{a_{2}},\dots,q^{a_{r}}\atop q% ^{b_{1}},q^{b_{2}},\dots,q^{b_{r}}};q,z\right)={{}_{r}H_{r}}\left({a_{1},a_{2}% ,\dots,a_{r}\atop b_{1},b_{2},\dots,b_{r}};z\right).$
##### 10: Bibliography D
• K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.
• A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
• J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
• T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.