Dalam matematika, **fungsi-H Fox** {\displaystyle H(x)} adalah sebuah perampatan dari fungsi-G Meijer dan fungsi Fox–Wright yang diperkenalkan oleh Charles Fox (1961). Fungsi tersebut didefinisikan oleh integral Mellin–Barnes


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{\displaystyle H\_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a\_{1},A\_{1})&(a\_{2},A\_{2})&\ldots &(a\_{p},A\_{p})\\(b\_{1},B\_{1})&(b\_{2},B\_{2})&\ldots &(b\_{q},B\_{q})\end{matrix}}\right.\right]={\frac {1}{2\pi i}}\int \_{L}{\frac {\prod \_{j=1}^{m}\Gamma (b\_{j}+B\_{j}s)\,\prod \_{j=1}^{n}\Gamma (1-a\_{j}-A\_{j}s)}{\prod \_{j=m+1}^{q}\Gamma (1-b\_{j}-B\_{j}s)\,\prod \_{j=n+1}^{p}\Gamma (a\_{j}+A\_{j}s)}}z^{-s}\,ds}

{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix}}\right.\right]={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}+B_{j}s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}-A_{j}s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-B_{j}s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}+A_{j}s)}}z^{-s}\,ds}
dimana {\displaystyle L} adalah kontur tertentu yang memisahkan kutub dari dua faktor dalam penyebut. Bandingkan dengan fungsi-G Meijer,

{\displaystyle G\_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a\_{1},\dots ,a\_{p}\\b\_{1},\dots ,b\_{q}\end{matrix}}\;\right|\,z\right)\,=\,{\frac {1}{2\pi i}}\int \_{L}{\frac {\prod \_{j=1}^{m}\Gamma (b\_{j}-s)\,\prod \_{j=1}^{n}\Gamma (1-a\_{j}+s)}{\prod \_{j=m+1}^{q}\Gamma (1-b\_{j}+s)\,\prod \_{j=n+1}^{p}\Gamma (a\_{j}-s)}}\,z^{s}\,ds}

Kasus khusus untuk yang H FOx dikurangi menjadi G Meijer adalah {\displaystyle A\_{j}=B\_{k}=C}, {\displaystyle C>0}, untuk {\displaystyle j=1\dots p} dan {\displaystyle k=1\dots q} ((Srivastava 1984, hlm. 50):


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{\displaystyle H\_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a\_{1},C)&(a\_{2},C)&\ldots &(a\_{p},C)\\(b\_{1},C)&(b\_{2},C)&\ldots &(b\_{q},C)\end{matrix}}\right.\right]={\frac {1}{C}}G\_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a\_{1},\dots ,a\_{p}\\b\_{1},\dots ,b\_{q}\end{matrix}}\;\right|\,z^{1/C}\right)}

{\displaystyle H_{p,q}^{\,m,n}\!\left[z\left|{\begin{matrix}(a_{1},C)&(a_{2},C)&\ldots &(a_{p},C)\\(b_{1},C)&(b_{2},C)&\ldots &(b_{q},C)\end{matrix}}\right.\right]={\frac {1}{C}}G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z^{1/C}\right)}
Sebuah perampatan dari fungsi-H Fox diberikan oleh Innayat Hussain (AA 1987). Untuk sebuah perampatan fungsi ini lebih lanjut, berguna dalam fisika dan statistika, lihat (Rathie 1997).

Referensi
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* Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", *Transactions of the American Mathematical Society*, **98**: 395–429, doi:10.2307/1993339, ISSN 0002-9947, JSTOR 1993339, MR 0131578
* Innayat-Hussain, AA (1987), "New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae", *J. Phys. A: Math. Gen.*, **20**: 4109–4117, doi:10.1088/0305-4470/20/13/019
* Innayat-Hussain, AA (1987), "New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function", *J. Phys. A: Math. Gen.*, **20**: 4119–4128, doi:10.1088/0305-4470/20/13/020
* Kilbas, Anatoly A. (2004), *H-Transforms: Theory and Applications*, CRC Press, ISBN 978-0415299169
* Mathai, A. M.; Saxena, Ram Kishore (1978), *The H-function with applications in statistics and other disciplines*, Halsted Press [John Wiley & Sons], New York-London-Sidney, ISBN 978-0-470-26380-8, MR 0513025
* Mathai, A. M.; Saxena, Ram Kishore; Haubold, Hans J. (2010), *The H-function*, Berlin, New York: Springer-Verlag, ISBN 978-1-4419-0915-2, MR 2562766
* Rathie, Arjun K. (1997), "A new generalization of generalized hypergeometric function", *Le Matematiche*, **LII**: 297–310 .
* Srivastava, H. M.; Gupta, K. C.; Goyal, S. P. (1982), *The H-functions of one and two variables*, New Delhi: South Asian Publishers Pvt. Ltd. MR 0691138
* Srivastava, H. M.; Manocha, H. L. (1984). *A treatise on generating functions*. ISBN 0-470-20010-3.

Pranala luar
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