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►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL, and spherical functions on certain nonsymmetric Gelfand pairs.
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►In his paper Lauricella’s hypergeometric function
(1963), he defined the -function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter.
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►Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval .
When , or , and are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.
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►Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval .
With the same conditions, and comprise a numerically satisfactory pair of solutions in the interval .
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►Then a pair of generators and can be chosen in an almost canonical way as follows.
…This yields a pair of generators that satisfy , , .
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►
(a)
In the general case, given by , we compute the roots ,
, , say, of the cubic equation ; see
§1.11(iii). These roots are necessarily distinct and represent ,
, in some order.
If and are real, and the discriminant is positive, that is ,
then , , can be identified via
(23.5.1), and , obtained from (23.6.16).
If , or and are not both real, then we label , , so
that the triangle with vertices , , is positively
oriented and is its longest side (chosen arbitrarily if
there is more than one). In particular, if , , are
collinear, then we label them so that is on the line segment
. In consequence, ,
satisfy
(with strict inequality unless
, , are collinear); also , .
Finally, on taking the principal square roots of and we obtain
values for and that lie in the 1st and 4th quadrants, respectively,
and , are given by
where denotes the arithmetic-geometric mean (see §§19.8(i) and
22.20(ii)). This process yields 2 possible pairs
(, ), corresponding to the 2 possible choices of the
square root.
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►In the following equations is any one of the four ordered pairs given in (10.63.1), and is either the same ordered pair or any other ordered pair in (10.63.1).
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►In consequence of (10.24.6), when is large and comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv).
Also, in consequence of (10.24.7)–(10.24.9), when is small either and or and comprise a numerically satisfactory pair depending whether or .
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►In consequence of (10.45.5)–(10.45.7), and comprise a numerically satisfactory pair of solutions of (10.45.1) when is large, and either and , or and , comprise a numerically satisfactory pair when is small, depending whether or .
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►Two solutions and are called a fundamental pair if any other solution is expressible as
…A fundamental pair can be obtained, for example, by taking any and requiring that
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►The following three statements are equivalent: and comprise a fundamental pair in ; does not vanish in ; and are linearly independent, that is, the only constants and such that
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►If is any one solution, and , are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as
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