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1: 19.2 Definitions
…
►where is a polynomial in while and are rational functions of .
…
►Here are real parameters, and and are real or complex variables, with , .
…
►If , then is pure imaginary.
…
►
§19.2(iv) A Related Function:
… ►For the special cases of and see (19.6.15). …2: 34.6 Definition: Symbol
3: 34.7 Basic Properties: Symbol
4: 34.5 Basic Properties: Symbol
5: 26.16 Multiset Permutations
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►Let be the multiset that has copies of , .
denotes the set of permutations of for all distinct orderings of the integers.
The number of elements in is the multinomial coefficient (§26.4) .
…
►The
-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
…and again with we have
…
6: 34.4 Definition: Symbol
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►
34.4.1
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►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1.
…
►
34.4.2
…
►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
7: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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►
is the number of ways of placing distinct objects into labeled boxes so that there are objects in the th box.
…
►These are given by the following equations in which are nonnegative integers such that
… is the multinominal coefficient (26.4.2):
…For each all possible values of are covered.
…
►where the summation is over all nonnegative integers such that .
…
8: 34.1 Special Notation
9: 34.3 Basic Properties: Symbol
…
►When any one of is equal to , or , the symbol has a simple algebraic form.
…For these and other results, and also cases in which any one of is or , see Edmonds (1974, pp. 125–127).
…
►Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
…
►See Srinivasa Rao and Rajeswari (1993, pp. 44–47) and references given there.
…
►For the polynomials see §18.3, and for the function see §14.30.
…
10: 34.2 Definition: Symbol
…
►The quantities in the symbol are called angular momenta.
…The corresponding projective quantum numbers
are given by
…
►
34.2.4
…
►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).