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1: 19.2 Definitions
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►Assume and , except that one of them may be 0, and .
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►The principal branch of and is , that is, the branch-cuts are .
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►Let .
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►If , then is pure imaginary.
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§19.2(iv) A Related Function:
…2: 10.55 Continued Fractions
3: 4.17 Special Values and Limits
4: 34.5 Basic Properties: Symbol
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►If any lower argument in a symbol is , , or , then the symbol has a simple algebraic form.
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34.5.4
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34.5.11
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34.5.18
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34.5.22
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5: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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is the multinominal coefficient (26.4.2):
… is the number of permutations of with cycles of length 1, cycles of length 2, , and cycles of length :
… is the number of set partitions of with subsets of size 1, subsets of size 2, , and subsets of size :
…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: 18.31 Bernstein–Szegő Polynomials
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►Let be a polynomial of degree and positive when .
The Bernstein–Szegő polynomials
, , are orthogonal on with respect to three types of weight function: , , .
In consequence, can be given explicitly in terms of and sines and cosines, provided that in the first case, in the second case, and in the third case.
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7: 24.20 Tables
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►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D.
►Wagstaff (1978) gives complete prime factorizations of and for and , respectively.
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►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).