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11: DLMF Project News
error generating summary12: 26.14 Permutations: Order Notation
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βΊThe set (§26.13) can be viewed as the collection of all ordered lists of elements of : .
As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
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βΊThe permutation has two descents: and .
…For example, .
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βΊIn this subsection is again the Stirling number of the second kind (§26.8), and is the th Bernoulli number (§24.2(i)).
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13: 25.6 Integer Arguments
14: Bibliography M
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Formulas and Theorems for the Special Functions of Mathematical Physics.
3rd edition, Springer-Verlag, New York-Berlin.
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On the Representation of Meijer’s -Function in the Vicinity of Singular Unity.
In Complex Analysis and Applications ’81 (Varna, 1981),
pp. 383–398.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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Lattice Statistics.
In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.),
University of California Engineering and Physical Sciences
Extension Series, pp. 96–143.
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15: Bibliography S
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Hypergeometric Functions and Their Applications.
Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.
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Decoherent dynamics of a two-level system coupled to a sea of spins.
Phys. Rev. Lett. 81 (26), pp. 5710–5713.
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Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities.
Math. Comp. 81 (280), pp. 2159–2173.
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On the calculation of complex zeros of the modified Bessel function of the second kind.
Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.
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Liouville-Green-Olver approximations for complex difference equations.
J. Approx. Theory 96 (2), pp. 301–322.
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16: Bibliography D
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The principal frequencies of vibrating systems with elliptic boundaries.
Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.
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Integralen voor de -functie van Riemann.
Mathematica (Zutphen) B5, pp. 170–180 (Dutch).
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Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält.
Abhandlungen der Königlich Preussischen Akademie der
Wissenschaften von 1837, pp. 45–81 (German).
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Orthogonal Polynomials of Several Variables.
Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge.
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Lamé instantons.
J. High Energy Phys. 2000 (1), pp. Paper 19, 8.
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17: Bibliography K
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A proof of the -Macdonald-Morris conjecture for
.
Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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Special functions and the Bieberbach conjecture.
Amer. Math. Monthly 95 (8), pp. 689–696.
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Elliptic asymptotics of the first and second Painlevé transcendents.
Uspekhi Mat. Nauk 49 (1(295)), pp. 77–140 (Russian).
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Dark optical solitons: Physics and applications.
Physics Reports 298 (2-3), pp. 81–197.
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The addition formula for Laguerre polynomials.
SIAM J. Math. Anal. 8 (3), pp. 535–540.
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18: 10.41 Asymptotic Expansions for Large Order
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βΊThe curve in the -plane is the upper boundary of the domain depicted in Figure 10.20.3 and rotated through an angle .
Thus is the point , where is given by (10.20.18).
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βΊThe series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large .
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βΊSimilar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20.
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βΊIt needs to be noted that the results (10.41.14) and (10.41.15) do not apply when or equivalently .
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19: 1.11 Zeros of Polynomials
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βΊSet to reduce to , with , .
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βΊ
, , , .
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βΊ
βΊ
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βΊResolvent cubic is with roots , , , and , , .
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20: 1.3 Determinants, Linear Operators, and Spectral Expansions
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βΊThe cofactor
of is
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βΊThe determinant of an upper or lower triangular, or diagonal, square matrix is the product of the diagonal elements .
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βΊLet
be defined for all integer values of and , and denote the determinant
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βΊSquare matices can be seen as linear operators because for all and , the space of all -dimensional vectors.
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βΊThe adjoint of a matrix is the matrix such that for all .
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