# Rogers–Dougall very well-poised sum

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##### 1: 16.4 Argument Unity
It is very well-poised if it is well-poised and $a_{1}=b_{1}+1$. …
##### 2: 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. …
17.4.3 ${{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.$
17.4.5 $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\sum_{m,n\geq 0}\frac{\left(a;q% \right)_{m+n}\left(b;q\right)_{m}\left(b^{\prime};q\right)_{n}x^{m}y^{n}}{% \left(q;q\right)_{m}\left(q;q\right)_{n}\left(c;q\right)_{m+n}},$
The series (17.4.1) is said to be well-poised when $r=s$ and … The series (17.4.1) is said to be very-well-poised when $r=s$, (17.4.11) is satisfied, and …
##### 3: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions
###### §17.9(iv) Bibasic Series
17.9.19 $\sum_{n=0}^{\infty}\frac{\left(a;q^{2}\right)_{n}\left(b;q\right)_{n}}{\left(q% ^{2};q^{2}\right)_{n}\left(c;q\right)_{n}}z^{n}=\frac{\left(b;q\right)_{\infty% }\left(az;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{2}\right)% _{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n}\left(z;q^{2}\right)% _{n}b^{2n}}{\left(q;q\right)_{2n}\left(az;q^{2}\right)_{n}}+\frac{\left(b;q% \right)_{\infty}\left(azq;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}% \left(zq;q^{2}\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n% +1}\left(zq;q^{2}\right)_{n}b^{2n+1}}{\left(q;q\right)_{2n+1}\left(azq;q^{2}% \right)_{n}}.$
17.9.20 $\sum_{n=0}^{\infty}\frac{\left(a;q^{k}\right)_{n}\left(b;q\right)_{kn}z^{n}}{% \left(q^{k};q^{k}\right)_{n}\left(c;q\right)_{kn}}=\frac{\left(b;q\right)_{% \infty}\left(az;q^{k}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{k}% \right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{n}\left(z;q^{k}% \right)_{n}b^{n}}{\left(q;q\right)_{n}\left(az;q^{k}\right)_{n}},$ $k=1,2,3,\dots$.
##### 4: Bibliography M
• S. C. Milne (1985a) A $q$-analog of the ${}_{5}F_{4}(1)$ summation theorem for hypergeometric series well-poised in $\mathit{SU}(n)$ . Adv. in Math. 57 (1), pp. 14–33.
• S. C. Milne (1985d) A $q$-analog of hypergeometric series well-poised in $\mathit{SU}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.
• S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
• S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
• S. Moch, P. Uwer, and S. Weinzierl (2002) Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43 (6), pp. 3363–3386.
• ##### 5: 17.18 Methods of Computation
Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …
##### 6: 17.12 Bailey Pairs
17.12.1 $\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n},$
$\beta_{n}=\sum_{j=0}^{n}\alpha_{j}u_{n-j}v_{n+j},$
$\gamma_{n}=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}.$
The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …
##### 7: 17.14 Constant Term Identities
###### Rogers–Ramanujan Constant Term Identities
17.14.2 $\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1}q^{2};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z% ^{-1}q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ % coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q% ;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1}q;q\right% )_{\infty}}=\frac{H(q)}{\left(-q;q^{2}\right)_{\infty}},$
17.14.3 $\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z^{-1}% q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ coeff.% of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1};q\right)_{% \infty}}=\frac{G(q)}{\left(-q;q^{2}\right)_{\infty}},$
17.14.4 $\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{2}% \right)_{n}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{% \infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{% \left(-z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(z^{-1};q% ^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}\mbox{ coeff. of % }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-2};q^{4}\right)_{% \infty}}=\frac{G(q^{4})}{\left(q;q^{2}\right)_{\infty}},$
17.14.5 $\sum_{n=0}^{\infty}\frac{q^{n^{2}+2n}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{% 2}\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right% )_{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty% }}{\left(-q^{2}z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(% z^{-1}q^{2};q^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}% \mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-% z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(q^{4}z^{% -2};q^{4}\right)_{\infty}}=\frac{H(q^{4})}{\left(q;q^{2}\right)_{\infty}}.$
##### 8: David M. Bressoud
His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …
##### 9: 26.10 Integer Partitions: Other Restrictions
where the last right-hand side is the sum over $m\geq 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … where the inner sum is the sum of all positive odd divisors of $t$. … where the inner sum is the sum of all positive divisors of $t$ that are in $S$.
###### §26.10(iv) Identities
Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …
##### 10: 18.33 Polynomials Orthogonal on the Unit Circle
When $a=0$ the Askey case is also known as the Rogers–Szegő case. See for a more general class Costa et al. (2012). …
18.33.21 $p(z)=\sum_{k=0}^{n}c_{k}z^{k},$ $c_{n}\neq 0$,