§34.6 Definition:
Symbol
►The
symbol may be defined either in terms of
symbols or equivalently in terms of
symbols:
►
34.6.1
►
34.6.2
►The
symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments.
…
§34.2 Definition:
Symbol
►The quantities
in the
symbol are called
angular momenta.
…They therefore satisfy the
triangle conditions
…The corresponding
projective quantum numbers
are given by
…
►When both conditions are satisfied the
symbol can be expressed as the finite sum
…
§34.4 Definition:
Symbol
►The
symbol is defined by the following double sum of products of
symbols:
…where the summation is taken over all admissible values of the
’s and
’s for each of the four
symbols; compare (
34.2.2) and (
34.2.3).
…
►The
symbol can be expressed as the finite sum
…
►where
is defined as in §
16.2.
…
§27.9 Quadratic Characters
►For an odd prime
, the
Legendre symbol
is defined as follows.
If
divides
, then the value of
is
.
…
►If an odd integer
has prime factorization
, then the
Jacobi symbol
is defined by
, with
.
The Jacobi
symbol
is a Dirichlet character (mod
).
…
§34.14 Tables
…
►Some selected
symbols are also given.
Other tabulations for
symbols are listed on pp.
11-12; for
symbols on pp.
16-17; for
symbols on p.
…
§34.10 Zeros
►In a
symbol, if the three angular momenta
do not satisfy the triangle conditions (
34.2.1), or if the projective quantum numbers do not satisfy (
34.2.3), then the
symbol is zero.
Similarly the
symbol (
34.4.1) vanishes when the triangle conditions are not satisfied by any of the four
symbols in the summation.
…However, the
and
symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled.
Such zeros are called
nontrivial zeros.
…
§34.12 Physical Applications
►The angular momentum coupling coefficients (
,
, and
symbols) are essential in the fields of nuclear, atomic, and molecular physics.
…
, and
symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see
Carlson and Rushbrooke (1950) and
Judd (1976).
§34.13 Methods of Computation
►Methods of computation for
and
symbols include recursion relations, see
Schulten and Gordon (1975a),
Luscombe and Luban (1998), and
Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these
symbols, see
Varshalovich et al. (1988, §§8.2.6, 9.2.1) and
Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these
symbols, see
Srinivasa Rao and Venkatesh (1978) and
Srinivasa Rao (1981).
►For
symbols, methods include evaluation of the single-sum series (
34.6.2), see
Fang and Shriner (1992); evaluation of triple-sum series, see
Varshalovich et al. (1988, §10.2.1) and
Srinivasa Rao et al. (1989).
…