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21: 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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22: 34.8 Approximations for Large Parameters
23: 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). …24: 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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25: 3.6 Linear Difference Equations
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►Given numerical values of and , the solution of the equation
…These errors have the effect of perturbing the solution by unwanted small multiples of and of an independent solution , say.
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►The unwanted multiples of now decay in comparison with , hence are of little consequence.
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►The latter method is usually superior when the true value of is zero or pathologically small.
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►beginning with .
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26: 10.75 Tables
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Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Wills et al. (1982) tabulates , , , for , 35D.
MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 10D; , , , 8D.
Achenbach (1986) tabulates , , , , , 19D or 19–21S.
27: 16.19 Identities
28: 17.5 Functions
29: 16.8 Differential Equations
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►is a value of at which all the coefficients , , are analytic.
If is not an ordinary point but , , are analytic at , then is a regular singularity.
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►where and are constants.
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►where indicates that the entry is omitted.
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►where indicates that the entry is omitted.
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30: 16.2 Definition and Analytic Properties
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►Throughout this chapter it is assumed that none of the bottom parameters , , , is a nonpositive integer, unless stated otherwise. Then formally
…Equivalently, the function is denoted by or , and sometimes, for brevity, by .
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►Suppose first one or more of the top parameters is a nonpositive integer.
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►See §16.5 for the definition of as a contour integral when and none of the is a nonpositive integer.
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►When and is fixed and not a branch point, any branch of is an entire function of each of the parameters .