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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: 34.2 Definition: Symbol
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►The quantities in the symbol are called angular momenta.
…The corresponding projective quantum numbers
are given by
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34.2.4
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►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
13: 16.12 Products
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: 5.10 Continued Fractions
18: 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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