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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: 34.1 Special Notation
6: 1.3 Determinants, Linear Operators, and Spectral Expansions
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►The cofactor
of is
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►For real-valued ,
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►where are the th roots of unity (1.11.21).
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►If tends to a limit as , then we say that the infinite determinant
converges and .
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►The corresponding eigenvectors can be chosen such that they form a complete orthonormal basis in .
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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: 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: 21.1 Special Notation
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►
►
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positive integers. | |
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th element of vector . | |
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Transpose of . | |
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. | |
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set of all elements of the form “”. | |
set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).) | |
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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.
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►where the summation is over all nonnegative integers such that .
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