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1: 19.2 Definitions
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►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: 34.7 Basic Properties: Symbol
4: 34.1 Special Notation
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►An often used alternative to the symbol is the Clebsch–Gordan coefficient
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34.1.1
►see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)).
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5: 26.9 Integer Partitions: Restricted Number and Part Size
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denotes the number of partitions of into at most parts.
See Table 26.9.1.
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►It follows that also equals the number of partitions of into parts that are less than or equal to .
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is the number of partitions of into at most parts, each less than or equal to .
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6: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
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►Note that .
…Note that .
►In the following examples, are the exponents in the factorization of in (27.2.1).
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►Table 27.2.1 lists the first 100 prime numbers .
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7: 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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8: 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.
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►where the summation is over all nonnegative integers such that .
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9: 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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