The Fox 
-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,](/images/equations/FoxH-Function/NumberedEquation1.svg) |
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.
The Fox 
-function
is implemented in the Wolfram Language
as FoxH.
A. Kilbas has derived a complete description for the asymptotic expansion of the 
-function.
Special cases of the Fox 
-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),](/images/equations/FoxH-Function/NumberedEquation2.svg) |
for 
,
..., 
,
, ..., 
complex number such that ![sum_(j=1)^(P+1)R[a_j]<sum_(j=1)^(P)Re[b_j]](/images/equations/FoxH-Function/Inline26.svg)
, 
, ![sgn(c)=sgn(I[z])](/images/equations/FoxH-Function/Inline28.svg)
, and 
is a generalized
hypergeometric function (Al-Musallam et al. 2001b).
-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 ![H_(pq)^(mn)[z|[a_p,A_p]; [b_p,B_p]]](/images/equations/FoxH-Function/Inline40.svg)
." §8.3 in 
-Function of One and Two Variables
with Applications. New Delhi, India: South Asian Publ., 1982.