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21: 23.10 Addition Theorems and Other Identities
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23.10.10
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23.10.17
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23.10.18
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23.10.19
►Also, when is replaced by the lattice invariants and are divided by and , respectively.
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22: 33.24 Tables
23: 18.36 Miscellaneous Polynomials
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§18.36(v) Non-Classical Laguerre Polynomials ,
… ►For the Laguerre polynomials this requires, omitting all strictly positive factors, … ►implying that, for , the orthogonality of the with respect to the Laguerre weight function , . …These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the polynomials, self-adjointness implying both orthogonality and completeness. … ►The resulting EOP’s, , satisfy …24: 23.2 Definitions and Periodic Properties
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►The generators of a given lattice are not unique.
…where are integers, then , are generators of iff
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►When the functions are related by
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►When it is important to display the lattice with the functions they are denoted by , , and , respectively.
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►If , is any pair of generators of , and is defined by (23.2.1), then
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25: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Derivatives and integrals with respect to the order of the Struve functions and
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J. Math. Anal. Appl. 137 (1), pp. 17–36.
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Note on the trivial zeros of Dirichlet -functions.
Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
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26: 18.41 Tables
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►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates , , , and for .
The ranges of are for and , and for and .
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►For , , and see §3.5(v).
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27: 19.33 Triaxial Ellipsoids
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►The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength , where is the demagnetizing factor, given in cgs units by
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19.33.7
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19.33.8
►where and are obtained from by permutation of , , and .
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28: 27.15 Chinese Remainder Theorem
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►Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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29: William P. Reinhardt
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►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.