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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: 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
…
βΊ
26.5.7
3: Bibliography H
…
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
Integrals containing the Fresnel functions and
.
Bul. Akad. Ε tiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).
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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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4: 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. …5: 19.2 Definitions
…
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). …6: 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 …7: 16.24 Physical Applications
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βΊ
§16.24(iii) , , and Symbols
… βΊThey can be expressed as functions with unit argument. …These are balanced functions with unit argument. Lastly, special cases of the symbols are functions with unit argument. …8: 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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9: 16.7 Relations to Other Functions
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βΊFor , , symbols see Chapter 34.
Further representations of special functions in terms of functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
10: 23 Weierstrass Elliptic and Modular
Functions
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