Fox–Wright function
In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function or just Wright function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on an idea of E. Maitland Wright (1935):
![{}_p\Psi_q \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}
; z \right]
=
\sum_{n=0}^\infty \frac{\Gamma( a_1 + A_1 n )\cdots\Gamma( a_p + A_p n )}{\Gamma( b_1 + B_1 n )\cdots\Gamma( b_q + B_q n )} \, \frac {z^n} {n!}.](../I/m/e7206e625879ffd25de09128bbac0f63.png)
Its normalisation
![{}_p\Psi^*_q \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}
; z \right]
=
\frac{ \Gamma(b_1) \cdots \Gamma(b_q) }{ \Gamma(a_1) \cdots \Gamma(a_p) }
\sum_{n=0}^\infty \frac{\Gamma( a_1 + A_1 n )\cdots\Gamma( a_p + A_p n )}{\Gamma( b_1 + B_1 n )\cdots\Gamma( b_q + B_q n )} \, \frac {z^n} {n!}](../I/m/bd56f38f18918f1b9f80af351a420bac.png)
becomes pFq(z) for A1...p = B1...q = 1.
The Fox–Wright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50):
![{}_p\Psi_q \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}
; z \right]
=
H^{1,p}_{p,q+1} \left[ -z \left| \begin{matrix}
( 1-a_1 , A_1 ) & ( 1-a_2 , A_2 ) & \ldots & ( 1-a_p , A_p ) \\
(0,1) & (1- b_1 , B_1 ) & ( 1-b_2 , B_2 ) & \ldots & ( 1-b_q , B_q ) \end{matrix} \right. \right].](../I/m/06e98f923c36279cdd4b63917fd545f7.png)
References
- Wright, E. M. (1935). "The asymptotic expansion of the generalized hypergeometric function". Proc. London Math. Soc. 10 (4): 286–293. doi:10.1112/jlms/s1-10.40.286.
- Srivastava, H.M.; Manocha, H.L. (1984). A treatise on generating functions. ISBN 0-470-20010-3.
- Miller, A. R.; Moskowitz, I.S. (1995). "Reduction of a Class of Fox–Wright Psi Functions for Certain Rational Parameters". Computers Math. Applic. 30 (11): 73–82.