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Fox H-Function


The Fox H-function is a very general function defined by

 H(z)=H_(p,q)^(m,n)[z|(a_1,alpha_1),...,(a_p,alpha_p); (b_1,beta_1),...,(b_p,beta_p)] 
=1/(2pii)int_C(product_(j=1)^(m)Gamma(b_j-beta_is)product_(j=1)^(n)Gamma(1-a_j+alpha_js))/(product_(j=n+1)^(p)Gamma(a_j+alpha_js)product_(j=m+1)^(q)Gamma(1-b_j+beta_js))z^sds,

where 0<=m<=q, 0<=n<=p, alpha_j,beta_j>0, and a_j,b_j are complex numbers such that no pole of Gamma(b_j-beta_js) for j=1, 2, ..., m coincides with any pole of Gamma(1-a_j+alpha_js) for j=1, 2, ..., n (Prudnikov et al. 1990, p. 626). In addition C, is a contour in the complex s-plane from omega-iinfty to omega+iinfty such that (b_j+k)/beta_j and (a_j-1-k)/alpha_j lie to the right and left of C, respectively.

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

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

Special cases of the Fox H-function include

 H_(2P+1)^1(z^i,c|(0,i); (1-a_j,i)_(P+1),(b_j-i)_P)=(_(p+1)F_P((a)_(P+1);(b)_P;(-1)^Pz))/(isgn(I[z])product_(j=1)^(P+1)Gamma(1-a_j)product_(j=1)^(P)Gamma(b)_j),

for a_1, ..., a_(P+1), b_1, ..., b_P complex number such that sum_(j=1)^(P+1)R[a_j]<sum_(j=1)^(P)Re[b_j], |z|=1, sgn(c)=sgn(I[z]), and _pF_q(z) is a generalized hypergeometric function (Al-Musallam et al. 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. "H-Function with Complex Parameters I: Existence." Int. J. Math. Math. Sci. 25, 571-586, 2001a.Al-Musallam, F. A. and Tuan, V. K. "H-Function with Complex Parameters II: Evaluation." Int. J. Math. Math. Sci. 25, 727-743, 2001b.Buschman, R. G. "H-Functions of Two Variables, I." Indian J. Math. 20, 139-153, 1978.Buschman, R. G. "Analytic Domains for Multivariable H-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 H-Functions." SIAM J. Appl. Math. 33, 542-558, 1977.Fox, C. "The G and H-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 H-Function of Two Variables." Rocky Mtn. J. Math. 22, 1317-1327, 1992.Mathai, A. M. and Saxena, R. K. The H-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 H-Function H_(pq)^(mn)[z|[a_p,A_p]; [b_p,B_p]]." §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 H-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 Web Resource. https://mathworld.wolfram.com/FoxH-Function.html

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