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11: 1.12 Continued Fractions
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and are called the th (canonical) numerator and denominator respectively.
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is equivalent to if there is a sequence , ,
, such that … ►Define … ►The continued fraction converges when … ►Then the convergents satisfy …
, such that … ►Define … ►The continued fraction converges when … ►Then the convergents satisfy …
12: 16.12 Products
13: 34.1 Special Notation
14: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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►The generalized hypergeometric function with matrix argument , numerator parameters , and denominator parameters is
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§35.8(iii) Case
… ►Let . … ►Let ; one of the be a negative integer; , , , . … ►Again, let . …15: 16.1 Special Notation
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►The main functions treated in this chapter are the generalized hypergeometric function , the Appell (two-variable hypergeometric) functions , , , , and the Meijer -function .
Alternative notations are , , and for the generalized hypergeometric function, , , , , for the Appell functions, and for the Meijer -function.
nonnegative integers. | |
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real or complex parameters. | |
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vector . | |
vector . | |
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16: 16.18 Special Cases
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►The and functions introduced in Chapters 13 and 15, as well as the more general functions introduced in the present chapter, are all special cases of the Meijer -function.
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16.18.1
►As a corollary, special cases of the and functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer -function.
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17: 16.3 Derivatives and Contiguous Functions
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16.3.1
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16.3.3
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16.3.4
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►Two generalized hypergeometric functions are (generalized)
contiguous if they have the same pair of values of and , and corresponding parameters differ by integers.
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16.3.7
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18: 34.8 Approximations for Large Parameters
19: 17.4 Basic Hypergeometric Functions
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§17.4(i) Functions
… ►Here and elsewhere it is assumed that the do not take any of the values . … ►§17.4(ii) Functions
… ►Here and elsewhere the must not take any of the values , and the must not take any of the values . … ►For the function see §16.4(v). …20: 16.11 Asymptotic Expansions
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►For subsequent use we define two formal infinite series, and , as follows:
…and .
Explicit representations for the coefficients are given in Volkmer (2023).
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►In this subsection we assume that none of is a nonpositive integer.
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►Explicit representations for the coefficients are given in Volkmer and Wood (2014).
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