Gauss%E2%80%93Hermite%20formula
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11: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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§35.8(iii) Case
►Kummer Transformation
… ►Pfaff–Saalschütz Formula
… ►Thomae Transformation
… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions and of matrix argument. …12: Bibliography
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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, , and the Ladies Diary.
Amer. Math. Monthly 95 (7), pp. 585–608.
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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions and , ,
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ACM Trans. Math. Software 3 (1), pp. 93–95.
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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument.
ACM Trans. Math. Software 16 (2), pp. 178–182.
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Special Functions.
Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.
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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
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13: 15.9 Relations to Other Functions
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15.9.2
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15.9.3
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15.9.5
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15.9.7
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►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers.
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14: 31.7 Relations to Other Functions
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§31.7(i) Reductions to the Gauss Hypergeometric Function
… ►Other reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where . … ►
31.7.2
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31.7.3
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►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities , , and , where and are related to as in §19.2(ii).
15: Bibliography K
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A proof of the -Macdonald-Morris conjecture for
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Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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On the evaluation of the Gauss hypergeometric function.
C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.
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Linear convergence and the bisection algorithm.
Amer. Math. Monthly 93 (1), pp. 48–51.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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Zeros of exceptional Hermite polynomials.
J. Approx. Theory 200, pp. 28–39.
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16: 20.11 Generalizations and Analogs
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§20.11(i) Gauss Sum
►For relatively prime integers with and even, the Gauss sum is defined by … … ► … ►Similar identities can be constructed for , , and . …17: 16.8 Differential Equations
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►the function satisfies the differential equation
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►We have the connection formula
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►Analytical continuation formulas for near are given in Bühring (1987b) for the case , and in Bühring (1992) for the general case.
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16.8.10
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18: 18.20 Hahn Class: Explicit Representations
19: 35.7 Gaussian Hypergeometric Function of Matrix Argument
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Gauss Formula
… ►Reflection Formula
… ►Subject to the conditions (a)–(c), the function is the unique solution of each partial differential equation … ►Systems of partial differential equations for the (defined in §35.8) and functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). … ►
35.7.10
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