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1: 34.6 Definition: Symbol
§34.6 Definition: Symbol
βΊThe symbol may be defined either in terms of symbols or equivalently in terms of symbols: βΊ
34.6.1
βΊ
34.6.2
βΊThe symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments.
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2: 19.37 Tables
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βΊ
Functions and
… βΊFunctions and
βΊTabulated for , to 10D by Fettis and Caslin (1964). … βΊTabulated for , to 6D by Byrd and Friedman (1971), for , and to 8D by Abramowitz and Stegun (1964, Chapter 17), and for , to 9D by Zhang and Jin (1996, pp. 674–675). … βΊTabulated for , , to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). …3: 27.2 Functions
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βΊFunctions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…( is defined to be 0.)
…It can be expressed as a sum over all primes :
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βΊis the sum of the th powers of the divisors of , where the exponent can be real or complex.
Note that .
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4: 8.23 Statistical Applications
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βΊThe function and its normalization play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275).
In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).
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5: 26.6 Other Lattice Path Numbers
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βΊ
Delannoy Number
βΊ is the number of paths from to that are composed of directed line segments of the form , , or . … βΊ
26.6.12
βΊ
26.6.13
βΊ
26.6.14
6: Bibliography H
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βΊ
Integrals that contain a probability function of complicated arguments.
Bul. Akad. Ε tiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).
βΊ
Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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βΊ
Some properties and applications of the repeated integrals of the error function.
Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
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βΊ
Coulomb Wave Functions.
In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.),
pp. 408–465.
βΊ
Variable precision exponential function.
ACM Trans. Math. Software 12 (2), pp. 79–91.
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7: Bibliography
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βΊ
Some determinants of Bernoulli, Euler and related numbers.
Portugal. Math. 18, pp. 91–99.
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βΊ
Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument.
ACM Trans. Math. Software 16 (2), pp. 178–182.
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βΊ
On basic hypergeometric series, mock theta functions, and partitions. II.
Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.
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βΊ
Umbral calculus, Bailey chains, and pentagonal number theorems.
J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
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βΊ
Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
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9: 33.23 Methods of Computation
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βΊThe power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii and , respectively, and may be used to compute the regular and irregular solutions.
…Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.
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βΊThus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii.
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βΊCombined with the Wronskians (33.2.12), the values of , , and their derivatives can be extracted.
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βΊ (16) should be
).
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10: 26.13 Permutations: Cycle Notation
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βΊAn explicit representation of can be given by the matrix:
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βΊis in cycle notation.
…In consequence, (26.13.2) can also be written as .
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βΊA permutation that consists of a single cycle of length can be written as the composition of two-cycles (read from right to left):
…A permutation with cycle type can be written as a product of transpositions, and no fewer.
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