Fox H-function

Generalization of the Meijer G-function and the Fox–Wright function

In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral

H p , q m , n [ z | ( a 1 , A 1 ) ( a 2 , A 2 ) … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … ( b q , B q ) ] = 1 2 π i ∫ L ∏ j = 1 m Γ ( b j + B j s ) ∏ j = 1 n Γ ( 1 − a j − A j s ) ∏ j = m + 1 q Γ ( 1 − b j − B j s ) ∏ j = n + 1 p Γ ( a j + A j s ) z − s d s , {\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,}

where L is a certain contour separating the poles of the two factors in the numerator.

Plot of the Fox H function H((((a 1,α 1),...,(a n,α n)),((a n+1,α n+1),...,(a p,α p)),(((b 1,β 1),...,(b m,β m)),in ((b m+1,β m+1),...,(b q,β q))),z) with H(((),()),(((-1,⁠1/2⁠)),()),z)

Relation to other Functions

Lambert W-function

A relation of the Fox H-Function to the -1 branch of the Lambert W-function is given by

W − 1 ⁡ ( − α ⋅ z ) ¯ = { lim β → α − [ α 2 ⋅ ( ( α − β ) ⋅ z ) α β β ⋅ H 1 , 2 1 , 1 ⁡ ( ( α + β β , α β ) ( 0 , 1 ) , ( − α β , α − β β ) ∣ − ( ( α − β ) ⋅ z ) α β − 1 ) ] , for | z | < 1 e | α | lim β → α − [ α 2 ⋅ ( ( α − β ) ⋅ z ) − α β β ⋅ H 2 , 1 1 , 1 ⁡ ( ( 1 , 1 ) , ( β − α β , α − β β ) ( − α β , α β ) ∣ − ( ( α − β ) ⋅ z ) 1 − α β ) ] , otherwise {\displaystyle {\overline {\operatorname {W} _{-1}\left(-\alpha \cdot z\right)}}={\begin{cases}\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{\frac {\alpha }{\beta }}}{\beta }}\cdot \operatorname {H} _{1,\,2}^{1,\,1}\left({\begin{matrix}\left({\frac {\alpha +\beta }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\left(0,\,1\right),\,\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{{\frac {\alpha }{\beta }}-1}\right)\right],\,{\text{for}}\left|z\right|<{\frac {1}{e\left|\alpha \right|}}\\\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{-{\frac {\alpha }{\beta }}}}{\beta }}\cdot \operatorname {H} _{2,\,1}^{1,\,1}\left({\begin{matrix}\left(1,\,1\right),\,\left({\frac {\beta -\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{1-{\frac {\alpha }{\beta }}}\right)\right],\,{\text{otherwise}}\\\end{cases}}} where z ¯ {\displaystyle {\overline {z}}} is the complex conjugate of z {\displaystyle z} .[1]

Meijer G-function

Compare to the Meijer G-function

G p , q m , n ( a 1 , … , a p b 1 , … , b q | z ) = 1 2 π i ∫ L ∏ j = 1 m Γ ( b j − s ) ∏ j = 1 n Γ ( 1 − a j + s ) ∏ j = m + 1 q Γ ( 1 − b j + s ) ∏ j = n + 1 p Γ ( a j − s ) z s d s . {\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.}

The special case for which the Fox H reduces to the Meijer G is Aj = Bk = C, C > 0 for j = 1...p and k = 1...q :[2]

H p , q m , n [ z | ( a 1 , C ) ( a 2 , C ) … ( a p , C ) ( b 1 , C ) ( b 2 , C ) … ( b q , C ) ] = 1 C G p , q m , n ( a 1 , … , a p b 1 , … , b q | z 1 / C ) . {\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).}

A generalization of the Fox H-function was given by Ram Kishore Saxena.[3][4] A further generalization of this function, useful in physics and statistics, was provided by A.M. Mathai and Ram Kishore Saxena.[5][6]

References

  1. ^ Rathie and Ozelim, Pushpa Narayan and Luan Carlos de Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023.
  2. ^ (Srivastava & Manocha 1984, p. 50)
  3. ^ Mathai, A. M.; Saxena, R. K.; Saxena, Ram Kishore (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer. ISBN 978-0-387-06482-6.
  4. ^ Innayat-Hussain (1987a)
  5. ^ Mathai, A. M.; Saxena, Rajendra Kumar (1978). The H-function with Applications in Statistics and Other Disciplines. Wiley. ISBN 978-0-470-26380-8.
  6. ^ Rathie (1997)
  • Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", Transactions of the American Mathematical Society, 98 (3): 395–429, doi:10.2307/1993339, ISSN 0002-9947, JSTOR 1993339, MR 0131578
  • Innayat-Hussain, AA (1987a), "New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae", J. Phys. A: Math. Gen., 20 (13): 4109–4117, Bibcode:1987JPhA...20.4109I, doi:10.1088/0305-4470/20/13/019
  • Innayat-Hussain, AA (1987b), "New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function", J. Phys. A: Math. Gen., 20 (13): 4119–4128, Bibcode:1987JPhA...20.4119I, 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. E. Horwood. ISBN 0-470-20010-3.