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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: 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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5: 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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6: 28.6 Expansions for Small
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►Leading terms of the power series for and for are:
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►The coefficients of the power series of , and also , are the same until the terms in and , respectively.
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►Numerical values of the radii of convergence of the power series (28.6.1)–(28.6.14) for are given in Table 28.6.1.
Here for , for , and for and .
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§28.6(ii) Functions and
…7: 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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8: 5.10 Continued Fractions
9: 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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10: 10.75 Tables
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Wills et al. (1982) tabulates , , , for , 35D.
MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 10D; , , , 8D.
Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of , for , 29S.
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 7D; , , , 6D.