# Dixon 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: Peter A. Clarkson
Dixon and J. …
##### 7: 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.
• A. L. Dixon and W. L. Ferrar (1930) Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.
• J. M. Dixon, J. A. Tuszyński, and P. A. Clarkson (1997) From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics. Oxford University Press, Oxford.
• 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.
• ##### 8: 4.27 Sums
###### §4.27 Sums
For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).
##### 10: 4.41 Sums
###### §4.41 Sums
For sums of hyperbolic functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).