The Fox -function is a special function defined by a Mellin-Barnes integral with positive slopes and in the gamma function arguments,

where , , , and are complex numbers such that no pole of for , 2, ..., coincides with any pole of for , 2, ..., (Prudnikov et al. 1990, p. 626). The integration contour separates the poles at from those at , where is a nonnegative integer. A vertical contour from to is one choice when the first family lies to its right, the second to its left, and the integral converges. When every and equals 1, the integral reduces to the Meijer G-function with the same parameters and (Fox 1961).

The Fox -function is implemented in the Wolfram Language as FoxH.

Kilbas and Saigo (1999) give power and power-logarithmic series expansions of the Fox -function near zero and infinity.

Al-Musallam and Tuan (2001ab) use a generalization with complex slopes. Its Mellin-Barnes integral is taken upward on with factor . A top pair contributes to the numerator, and a bottom pair contributes to the denominator. For nonreal with , using the principal value of and taking with the sign of , they obtain

where and denote the lists of upper and lower parameters of the generalized hypergeometric function. The formula applies when and the displayed gamma and hypergeometric function factors are defined (Al-Musallam and Tuan 2001b).

See also

Kampe de Feriet Function, MacRobert's E-Function, Meijer G-Function

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References

Al-Musallam, F. A. and Tuan, V. K. "-Function with Complex Parameters I: Existence." Int. J. Math. Math. Sci. 25, 571-586, 2001a.Al-Musallam, F. A. and Tuan, V. K. "-Function with Complex Parameters II: Evaluation." Int. J. Math. Math. Sci. 25, 727-743, 2001b.Buschman, R. G. "-Functions of Two Variables, I." Indian J. Math. 20, 139-153, 1978.Buschman, R. G. "Analytic Domains for Multivariable -Functions." Pure Appl. Math. Sci. 16, 23-27, 1982.Carter, B. D. and Springer, M. D. "The Distribution of Products, Quotients, and Powers of Independent -Functions." SIAM J. Appl. Math. 33, 542-558, 1977.Fox, C. "The and -Functions as Symmetrical Fourier Kernels." Trans. Amer. Math. Soc. 98, 395-429, 1961.Hai, N.; Marichev, O. I.; and Buschman, R. G. "Theory of the General -Function of Two Variables." Rocky Mtn. J. Math. 22, 1317-1327, 1992.Kilbas, A. A. and Saigo, M. "On the -Function." J. Appl. Math. Stochastic Anal. 12, 191-204, 1999. https://doi.org/10.1155/S1048953399000192.Mathai, A. M. and Saxena, R. K. The H-Function with Applications in Statistics and Other Disciplines. New Delhi, India: Wiley, 1978.Nguyen Thanh Hai and Yakubovich, S. B. The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory. Singapore: World Scientific, 1992. https://doi.org/10.1142/1425.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. "Evaluation of Integrals and the Mellin Transform." Itogi Nauki i Tekhniki, Seriya Matemat. Analiz 27, 3-146, 1989.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Fox -Function ." §8.3 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 626-629, 1990.

Srivastava, H. M.; Gupta, K. C.; and Goyal, S. P. The -Function of One and Two Variables with Applications. New Delhi, India: South Asian Publ., 1982.

Yakubovich, S. B. and Luchko, Y. F. The Hypergeometric Approach to Integral Transforms and Convolutions. Amsterdam, Netherlands: Kluwer, 1994.

Referenced on Wolfram|Alpha

Fox H-Function

Cite this as:

Weisstein, Eric W. "Fox H-Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FoxH-Function.html

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