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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: 34.7 Basic Properties: Symbol
4: 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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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.
…
►where the summation is over all nonnegative integers such that .
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6: 34.1 Special Notation
7: 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 .
►
is the number of partitions of into at most parts, each less than or equal to .
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8: 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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9: 19.29 Reduction of General Elliptic Integrals
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►Let
…where
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►Next, for , define , and assume both ’s are positive for .
…where
…If , where both linear factors are positive for , and , then (19.29.25) is modified so that
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10: 3.6 Linear Difference Equations
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►Given numerical values of and , the solution of the equation
…These errors have the effect of perturbing the solution by unwanted small multiples of and of an independent solution , say.
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►The unwanted multiples of now decay in comparison with , hence are of little consequence.
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►The latter method is usually superior when the true value of is zero or pathologically small.
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►beginning with .
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