Lauricella%0Afunction
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1: 19.15 Advantages of Symmetry
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►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s
(Carlson (1961b)).
The function (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in , and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation.
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2: 19.25 Relations to Other Functions
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►Assume , , and .
…with .
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§19.25(vii) Hypergeometric Function
… ►For these results and extensions to the Appell function (§16.13) and Lauricella’s function see Carlson (1963). ( and are equivalent to the -function of 3 and variables, respectively, but lack full symmetry.) …3: Bille C. Carlson
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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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4: Bibliography C
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Lauricella’s hypergeometric function
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J. Math. Anal. Appl. 7 (3), pp. 452–470.
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A Fortran subroutine for the Bessel function of order to
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Comput. Phys. Comm. 21 (1), pp. 109–118.
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5: 33.24 Tables
6: 26.15 Permutations: Matrix Notation
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►The set (§26.13) can be identified with the set of matrices of 0’s and 1’s with exactly one 1 in each row and column.
The permutation corresponds to the matrix in which there is a 1 at the intersection of row with column , and 0’s in all other positions.
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►Define .
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►The Ferrers board of shape , , is the set .
…If is the Ferrers board of shape , then
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7: 24.2 Definitions and Generating Functions
8: 11.14 Tables
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Abramowitz and Stegun (1964, Chapter 12) tabulates , , and for and , to 6D or 7D.
Agrest et al. (1982) tabulates and for and to 11D.
Abramowitz and Stegun (1964, Chapter 12) tabulates and for to 5D or 7D; , , and for to 6D.
Agrest et al. (1982) tabulates and for to 11D.
Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function for , , and , together with surface plots.