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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: 19.2 Definitions
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βΊThe principal values of and are even functions.
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βΊ
§19.2(iv) A Related Function:
… βΊFormulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using . … βΊWhen and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if : …For the special cases of and see (19.6.15). …3: 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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4: 1.12 Continued Fractions
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βΊ
is called the th approximant or convergent to
.
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βΊDefine
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βΊA contraction of a continued fraction is a continued fraction whose convergents form a subsequence of the convergents of .
Conversely, is called an extension of .
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βΊThen the convergents satisfy
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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 … βΊ …6: 26.5 Lattice Paths: Catalan Numbers
…
βΊ
is the Catalan number.
…(Sixty-six equivalent definitions of are given in Stanley (1999, pp. 219–229).)
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βΊ
26.5.3
βΊ
26.5.4
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βΊ
26.5.7
7: 34.14 Tables
§34.14 Tables
βΊTables of exact values of the squares of the and symbols in which all parameters are are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of , and symbols on pp. … βΊSome selected symbols are also given. … 16-17; for symbols on p. … βΊ 310–332, and for the symbols on pp. …8: Bibliography H
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βΊ
Integrals containing the Fresnel functions and
.
Bul. Akad. Ε tiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).
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βΊ
Lamé polynomials of large order.
SIAM J. Math. Anal. 8 (5), pp. 800–842.
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βΊ
Algorithm 56: Complete elliptic integral of the second kind.
Comm. ACM 4 (4), pp. 180–181.
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βΊ
Calculation of Coulomb wave functions for high energies.
Phys. Rev. 54 (8), pp. 627–628.
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βΊ
On congruences involving Bernoulli numbers and irregular primes. II.
Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
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9: Bibliography K
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βΊ
A proof of the -Macdonald-Morris conjecture for
.
Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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βΊ
Poly-Bernoulli numbers.
J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.
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βΊ
Replica field theories, Painlevé transcendents, and exact correlation functions.
Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).
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βΊ
Askey-Wilson Polynomials for Root Systems of Type
.
In Hypergeometric Functions on Domains of Positivity, Jack
Polynomials, and Applications (Tampa, FL, 1991),
Contemp. Math., Vol. 138, pp. 189–204.
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βΊ
On the zeros of the Fresnel integrals.
Canad. J. Math. 9, pp. 118–131.
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10: 27.2 Functions
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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 .
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βΊ