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Fox H-Function

المؤلف:  Al-Musallam, F. A. and Tuan, V. K

المصدر:  "H-Function with Complex Parameters I: Existence." Int. J. Math. Math. Sci. 25,

الجزء والصفحة:  ...

15-6-2019

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Fox H-Function

The Fox -function is a very general function defined by

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). In addition , is a contour in the complex -plane from  to  such that  and  lie to the right and left of , respectively.

A. Kilbas has derived a complete description for the asymptotic expansion of the -function.

Special cases of the Fox -function include

for , ..., , , ...,  complex number such that , , , and  is a generalized hypergeometric function (Al-Musallam et al. 2001b).

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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.

Mathai, A. M. and Saxena, R. K. The -Function with Applications in Statistics and Other Disciplines.0470263806 New Delhi, India: Wiley, 1978.

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.