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finite sum of 3j symbols

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1: 34.6 Definition: 9 j Symbol
The 9 j symbol may be defined either in terms of 3 j symbols or equivalently in terms of 6 j symbols:
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  m r s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
2: 34.4 Definition: 6 j Symbol
§34.4 Definition: 6 j Symbol
For alternative expressions for the 6 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 3 4 of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
3: 34.2 Definition: 3 j Symbol
When both conditions are satisfied the 3 j symbol can be expressed as the finite sumwhere F 2 3 is defined as in §16.2. For alternative expressions for the 3 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 2 3 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
4: 34.5 Basic Properties: 6 j Symbol
§34.5 Basic Properties: 6 j Symbol
Examples are provided by: …
§34.5(ii) Symmetry
§34.5(vi) Sums
They constitute addition theorems for the 6 j symbol. …
5: Bibliography R
  • C. C. J. Roothaan and S. Lai (1997) Calculation of 3 n - 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 9 j -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 α = m = 1 , 2 , 3 , the left-hand side of (10.22.36) is the m th repeated integral of J ν ( x ) (§§1.4(v) and 1.15(vi)). … where j ν , and j ν , m are zeros of J ν ( x ) 10.21(i)), and δ , m is Kronecker’s symbol. … Equation (10.22.70) also remains valid if the order ν + 1 of the J functions on both sides is replaced by ν + 2 n 3 , n = 1 , 2 , , and the constraint ν > 3 2 is replaced by ν > n + 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). …
    complex plane (excluding infinity).
    < is finite, or converges.
    ( a , b ] or [ a , b ) half-closed intervals.
    ( α ) n Pochhammer’s symbol: α ( α + 1 ) ( α + 2 ) ( α + n 1 ) if n = 1 , 2 , 3 , ; 1 if n = 0 .
    For example, to 4D π is 3.1415 (unrounded) and 3. …  J. …
    8: 18.28 Askey–Wilson Class
    The Askey–Wilson polynomials form a system of OP’s { p n ( x ) } , n = 0 , 1 , 2 , , that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. …
    18.28.1 p n ( x ) = p n ( x ; a , b , c , d | q ) = a n = 0 n q ( a b q , a c q , a d q ; q ) n ( q n , a b c d q n 1 ; q ) ( q ; q ) j = 0 1 ( 1 2 a q j x + a 2 q 2 j ) ,
    Also, x are the points 1 2 ( α q + α 1 q ) with α any of the a , b , c , d whose absolute value exceeds 1 , and the sum is over the = 0 , 1 , 2 , with | α q | > 1 . …
    18.28.7 Q n ( cos θ ; a , b | q ) = p n ( cos θ ; a , b , 0 , 0 | q ) = a n = 0 n q ( a b q ; q ) n ( q n ; q ) ( q ; q ) j = 0 1 ( 1 2 a q j cos θ + a 2 q 2 j ) = ( a b ; q ) n a n ϕ 2 3 ( q n , a e i θ , a e i θ a b , 0 ; q , q ) = ( b e i θ ; q ) n e i n θ ϕ 1 2 ( q n , a e i θ b 1 q 1 n e i θ ; q , b 1 q e i θ ) .
    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 y = 0 N Q n ( q y ) Q m ( q y ) [ N y ] q ( α q ; q ) y ( β q ; q ) N y ( α q ) y = h n δ n , m , n , m = 0 , 1 , , N ,
    18.27.14 y = 0 p n ( q y ) p m ( q y ) ( b q ; q ) y ( a q ) y ( q ; q ) y = h n δ n , m , 0 < a < q 1 , b < q 1 ,
    18.27.22 = 0 ( h n ( q ; q ) h m ( q ; q ) + h n ( q ; q ) h m ( q ; q ) ) ( q + 1 , q + 1 ; q ) q = ( q ; q ) n ( q , 1 , q ; q ) q n ( n 1 ) / 2 δ n , m .
    18.27.24 = ( h ~ n ( c q ; q ) h ~ m ( c q ; q ) + h ~ n ( c q ; q ) h ~ m ( c q ; q ) ) q ( c 2 q 2 ; q 2 ) = 2 ( q 2 , c 2 q , c 2 q ; q 2 ) ( q , c 2 , c 2 q 2 ; q 2 ) ( q ; q ) n q n 2 δ n , m , c > 0 .
    10: 18.30 Associated OP’s
    18.30.5 ( 1 ) n ( α + β + c + 1 ) n n ! P n ( α , β ) ( x ; c ) ( α + β + 2 c + 1 ) n ( β + c + 1 ) n = = 0 n ( n ) ( n + α + β + 2 c + 1 ) ( c + 1 ) ( β + c + 1 ) ( 1 2 x + 1 2 ) F 3 4 ( n , n + + α + β + 2 c + 1 , β + c , c β + + c + 1 , + c + 1 , α + β + 2 c ; 1 ) ,
    where the generalized hypergeometric function F 3 4 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. …