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11: 24.2 Definitions and Generating Functions
§24.2 Definitions and Generating Functions
►§24.2(i) Bernoulli Numbers and Polynomials
… ►§24.2(ii) Euler Numbers and Polynomials
… ► …12: 26.6 Other Lattice Path Numbers
§26.6 Other Lattice Path Numbers
… ►Delannoy Number
… ►Motzkin Number
… ►Narayana Number
… ►§26.6(iv) Identities
…13: 26.2 Basic Definitions
14: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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A Classical Introduction to Modern Number Theory.
2nd edition, Springer-Verlag, New York.
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15: 5.11 Asymptotic Expansions
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5.11.1
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5.11.2
►For the Bernoulli numbers
, see §24.2(i).
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►Wrench (1968) gives exact values of up to .
…For explicit formulas for in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of as see Boyd (1994) and Nemes (2015a).
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16: Peter L. Walker
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►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan.
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17: Software Index
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Software Associated with Books.
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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24.21(ii) , , , | ✓ | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Derive, MuPAD | ||||||||||||
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27 Functions of Number Theory | |||||||||||||||||||||||||
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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.
18: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into distinct parts.
denotes the number of partitions of into at most distinct parts.
denotes the number of partitions of into parts with difference at least .
… denotes the number of partitions of into odd parts.
denotes the number of partitions of into parts taken from the set .
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19: 26.12 Plane Partitions
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►Then the number of plane partitions in is
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►The number of symmetric plane partitions in is
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►The number of cyclically symmetric plane partitions in is
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►The number of descending plane partitions in is
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20: 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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Congruences of -adic integer order Bernoulli numbers.
J. Number Theory 59 (2), pp. 374–388.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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A Centennial History of the Prime Number Theorem.
In Number Theory,
Trends Math., pp. 1–14.
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