sigma%20function
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1: William P. Reinhardt
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►He has been a National Lecturer for Sigma Xi and Phi Beta Kappa, as well as a Sloan, Dreyfus, and Guggenheim Fellow, and Fulbright Senior Scholar (Australia).
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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.
2: 27.2 Functions
§27.2 Functions
… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►Note that . …This is Jordan’s function. … ►Table 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for . …3: 26.14 Permutations: Order Notation
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►The set (§26.13) can be viewed as the collection of all ordered lists of elements of : .
…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 major index is the sum of all positions that mark the first element of a descent:
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►An excedance in is a position for which .
A weak excedance is a position for which .
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4: 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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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Uniform asymptotic expansion for the incomplete beta function.
SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
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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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Hypergeometric functions.
Acta Math. 94, pp. 289–349.
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5: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators.
SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
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6: 26.2 Basic Definitions
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►Thus is the permutation , , .
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►Given a finite set with permutation , a cycle is an ordered equivalence class of elements of where is equivalent to if there exists an such that , where and is the composition of with .
It is ordered so that follows .
…Here , and .
The function
also interchanges 3 and 6, and sends 4 to itself.
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7: 23.9 Laurent and Other Power Series
8: 26.12 Plane Partitions
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§26.12(ii) Generating Functions
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26.12.25
►where is the sum of the squares of the divisors of .
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26.12.26
►where is the Riemann -function (§25.2(i)).
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9: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Gauss hypergeometric representations of the Ferrers function of the second kind.
SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.
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On parameter differentiation for integral representations of associated Legendre functions.
SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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10: 26.13 Permutations: Cycle Notation
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is a one-to-one and onto mapping from to itself.
An explicit representation of can be given by the matrix:
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►In cycle notation, the elements in each cycle are put inside parentheses, ordered so that immediately follows or, if is the last listed element of the cycle, then is the first element of the cycle.
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►See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►Given a permutation , the inversion number of , denoted , is the least number of adjacent transpositions required to represent .
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