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1: 19.2 Definitions
…
►where is a polynomial in while and are rational functions of .
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►Here are real parameters, and and are real or complex variables, with , .
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►If , then is pure imaginary.
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§19.2(iv) A Related Function:
… ►For the special cases of and see (19.6.15). …2: 34.6 Definition: Symbol
3: 34.5 Basic Properties: Symbol
4: 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.
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34.4.2
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►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).
5: 34.7 Basic Properties: Symbol
6: 34.3 Basic Properties: Symbol
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►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).
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►Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
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34.3.8
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►For the polynomials see §18.3, and for the function see §14.30.
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7: 16.10 Expansions in Series of Functions
§16.10 Expansions in Series of Functions
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16.10.1
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16.10.2
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►Expansions of the form are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).
8: 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) .
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►The
-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
…and again with we have
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9: 34.2 Definition: Symbol
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►The quantities in the symbol are called angular momenta.
…The corresponding projective quantum numbers
are given by
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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).
10: 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.
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►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 .
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