# finite sum of 3j symbols

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##### 1: 34.6 Definition: $\mathit{9j}$ Symbol
The $\mathit{9j}$ symbol may be defined either in terms of $\mathit{3j}$ symbols or equivalently in terms of $\mathit{6j}$ symbols:
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
##### 2: 34.4 Definition: $\mathit{6j}$ Symbol
###### §34.4 Definition: $\mathit{6j}$Symbol
For alternative expressions for the $\mathit{6j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{4}F_{3}}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
##### 3: 34.2 Definition: $\mathit{3j}$ Symbol
When both conditions are satisfied the $\mathit{3j}$ symbol can be expressed as the finite sumwhere ${{}_{3}F_{2}}$ is defined as in §16.2. For alternative expressions for the $\mathit{3j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{3}F_{2}}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
##### 4: 34.5 Basic Properties: $\mathit{6j}$ Symbol
###### §34.5 Basic Properties: $\mathit{6j}$Symbol
Examples are provided by: …
###### §34.5(vi) Sums
They constitute addition theorems for the $\mathit{6j}$ symbol. …
##### 5: Bibliography R
• C. C. J. Roothaan and S. Lai (1997) Calculation of $3n$-$j$ symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.
• H. Rosengren (1999) Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
• G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.
• M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr. (1959) The $3$-$j$ and $6$-$j$ Symbols. The Technology Press, MIT, Cambridge, MA.
• K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
• ##### 6: 10.22 Integrals
###### §10.22(ii) Integrals over Finite Intervals
When $\alpha=m=1,2,3,\ldots$ the left-hand side of (10.22.36) is the $m$th repeated integral of $J_{\nu}\left(x\right)$ (§§1.4(v) and 1.15(vi)). … where $j_{\nu,\ell}$ and $j_{\nu,m}$ are zeros of $J_{\nu}\left(x\right)$10.21(i)), and $\delta_{\ell,m}$ is Kronecker’s symbol. … Equation (10.22.70) also remains valid if the order $\nu+1$ of the $J$ functions on both sides is replaced by $\nu+2n-3$, $n=1,2,\dots$, and the constraint $\Re\nu>-\frac{3}{2}$ is replaced by $\Re\nu>-n+\frac{1}{2}$. …
##### 7: Mathematical Introduction
These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …
 $\mathbb{C}$ complex plane (excluding infinity). … is finite, or converges. …
 $(a,b]$ or $[a,b)$ half-closed intervals. … Pochhammer’s symbol: $\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)$ if $n=1,2,3,\dotsc$; 1 if $n=0$. …
For example, to 4D $\pi$ is $3.1415\ldots$ (unrounded) and 3. …  J. …
The Askey–Wilson polynomials form a system of OP’s $\{p_{n}(x)\}$, $n=0,1,2,\dots$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. … Also, $x_{\ell}$ are the points $\tfrac{1}{2}(\alpha q^{\ell}+\alpha^{-1}q^{-\ell})$ with $\alpha$ any of the $a,b,c,d$ whose absolute value exceeds $1$, and the sum is over the $\ell=0,1,2,\dots$ with $|\alpha q^{\ell}|>1$. …
18.28.7 $Q_{n}\left(\cos\theta;a,b\,|\,q\right)=p_{n}\left(\cos\theta;a,b,0,0\,|\,q% \right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(abq^{\ell};q\right)_{n-\ell% }\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(% 1-2aq^{j}\cos\theta+a^{2}q^{2j})=\frac{\left(ab;q\right)_{n}}{a^{n}}{{}_{3}% \phi_{2}}\left({q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta},a{\mathrm{e}}^{-% \mathrm{i}\theta}\atop ab,0};q,q\right)=\left(b{\mathrm{e}}^{-\mathrm{i}\theta% };q\right)_{n}{\mathrm{e}}^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},a{% \mathrm{e}}^{\mathrm{i}\theta}\atop b^{-1}q^{1-n}{\mathrm{e}}^{\mathrm{i}% \theta}};q,b^{-1}q{\mathrm{e}}^{-\mathrm{i}\theta}\right).$
Leonard (1982) classified all (finite or infinite) discrete systems of OP’s $p_{n}(x)$ on a set $\{x(m)\}$ for which there is a system of discrete OP’s $q_{m}(y)$ on a set $\{y(n)\}$ such that $p_{n}(x(m))=q_{m}(y(n))$. …
##### 9: 18.27 $q$-Hahn Class
Here $a,b$ are fixed positive real numbers, and $I_{+}$ and $I_{-}$ are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. …
18.27.4 $\sum_{y=0}^{N}Q_{n}(q^{-y})Q_{m}(q^{-y})\genfrac{[}{]}{0.0pt}{}{N}{y}_{q}\frac% {\left(\alpha q;q\right)_{y}\left(\beta q;q\right)_{N-y}}{\left(\alpha q\right% )^{y}}=h_{n}\delta_{n,m},$ $n,m=0,1,\ldots,N$,
18.27.14 $\sum_{y=0}^{\infty}p_{n}(q^{y})p_{m}(q^{y})\frac{\left(bq;q\right)_{y}(aq)^{y}% }{\left(q;q\right)_{y}}=h_{n}\delta_{n,m},$ $0,
18.27.22 $\sum_{\ell=0}^{\infty}\left(h_{n}\left(q^{\ell};q\right)h_{m}\left(q^{\ell};q% \right)+h_{n}\left(-q^{\ell};q\right)h_{m}\left(-q^{\ell};q\right)\right)\*% \left(q^{\ell+1},-q^{\ell+1};q\right)_{\infty}q^{\ell}=\left(q;q\right)_{n}% \left({q,-1,-q};q\right)_{\infty}q^{n(n-1)/2}\delta_{n,m}.$
18.27.24 $\sum_{\ell=-\infty}^{\infty}\left(\tilde{h}_{n}\left(cq^{\ell};q\right)\tilde{% h}_{m}\left(cq^{\ell};q\right)+\tilde{h}_{n}\left(-cq^{\ell};q\right)\tilde{h}% _{m}\left(-cq^{\ell};q\right)\right)\frac{q^{\ell}}{\left(-c^{2}q^{2\ell};q^{2% }\right)_{\infty}}=2\frac{\left(q^{2},-c^{2}q,-c^{-2}q;q^{2}\right)_{\infty}}{% \left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q\right)_{n}}{% q^{n^{2}}}\delta_{n,m},$ $c>0$.
18.30.5 $\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n!\,P^{(\alpha,\beta)}_{n}% \left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}}{\left(\beta+c+1\right)_% {n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}{\left(n+\alpha+\beta+2c+% 1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(\beta+c+1\right)_{\ell}}}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{4}F_{3}}\left({\ell-n,n+% \ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,\ell+c+1,\alpha+\beta+2c};% 1\right),$
where the generalized hypergeometric function ${{}_{4}F_{3}}$ is defined by (16.2.1). … For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). … They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10). … The type 3 Pollaczek polynomials are the associated type 2 Pollaczek polynomials, see §18.35. …