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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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βΊTabulated for , to 10D by Fettis and Caslin (1964).
βΊTabulated for , to 7S by BeliΝ‘akov et al. (1962).
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βΊTabulated for , to 10D by Fettis and Caslin (1964).
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βΊTabulated (with different notation) for , , to 5D by Abramowitz and Stegun (1964, Chapter 17), and for , , to 7D by Zhang and Jin (1996, pp. 676–677).
βΊTabulated for , , to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)).
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3: 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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4: 27.2 Functions
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βΊ
27.2.9
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βΊIt is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…Note that .
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βΊTable 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
βΊ
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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 … βΊ
26.6.4
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βΊ
26.6.10
,
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6: Bibliography
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βΊ
A harmonic mean inequality for the gamma function.
J. Comput. Appl. Math. 87 (2), pp. 195–198.
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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
A trinomial analogue of Bailey’s lemma and superconformal invariance.
Comm. Math. Phys. 192 (2), pp. 245–260.
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βΊ
Bernoulli’s power-sum formulas revisited.
Math. Gaz. 90 (518), pp. 276–279.
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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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7: 22.7 Landen Transformations
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8: Bibliography H
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βΊ
The Laplace transform for expressions that contain a probability function.
Bul. Akad. Ε tiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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βΊ
Expansions for the probability function in series of ΔebyΕ‘ev polynomials and Bessel functions.
Bul. Akad. Ε tiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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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9: 3.4 Differentiation
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βΊThe are the differentiated Lagrangian interpolation coefficients:
…where is as in (3.3.10).
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βΊ
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βΊwhere is a simple closed contour described in the positive rotational sense such that and its interior lie in the domain of analyticity of , and is interior to .
Taking to be a circle of radius centered at , we obtain
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10: 10.31 Power Series
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