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1: 25.6 Integer Arguments
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§25.6(i) Function Values
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25.6.2
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25.6.3
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25.6.11
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►For related results see Basu and Apostol (2000).
2: 26.14 Permutations: Order Notation
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►Equivalently, this is the sum over of the number of integers less than that lie in positions to the right of the th position:
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►The Eulerian number is equal to the number of permutations in with exactly excedances.
It is also equal to the number of permutations in with exactly weak excedances.
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3: 25.12 Polylogarithms
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►The remainder of the equations in this subsection apply to principal branches.
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►The special case is the Riemann zeta function: .
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►Further properties include
…and
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►In terms of polylogarithms
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4: 26.6 Other Lattice Path Numbers
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Delannoy Number
… ►Motzkin Number
… ►Narayana Number
… ►Schröder Number
… ►§26.6(iii) Recurrence Relations
…5: William P. Reinhardt
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►Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland.
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►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions.
Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions.
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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.
6: 27.2 Functions
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►( is defined to be 0.)
Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
…They tend to thin out among the large integers, but this thinning out is not completely regular.
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►the sum of the th powers of the positive integers that are relatively prime to
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►is the number of -tuples of integers whose greatest common divisor is relatively prime to
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7: 6.16 Mathematical Applications
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►Hence, if is fixed and , then , , or according as , , or ; compare (6.2.14).
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►Hence if and , then the limiting value of overshoots by approximately 18%.
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►If we assume Riemann’s hypothesis that all nonreal zeros of have real part of (§25.10(i)), then
…where is the number of primes less than or equal to
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8: 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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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Integral equations and relations for Lamé functions.
Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
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9: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors.
2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
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Confluent hypergeometric equations and related solvable potentials in quantum mechanics.
J. Math. Phys. 41 (12), pp. 7964–7996.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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10: 28.8 Asymptotic Expansions for Large
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►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3).
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►Then as
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►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5).
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►Then as
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►For related results see Langer (1934) and Sharples (1967, 1971).
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