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 _pF_q(z) based on an idea of E. Maitland Wright (1935):

${}_p\Psi_q \left[\begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} ; z \right] = \sum_{n=0}^\infty \frac{\Gamma( a_1 + A_1 n )\cdots\Gamma( a_p + A_p n )}{\Gamma( b_1 + B_1 n )\cdots\Gamma( b_q + B_q n )} \, \frac {z^n} {n!}.$

Its normalisation

${}_p\Psi^*_q \left[\begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} ; z \right] = \frac{ \Gamma(b_1) \cdots \Gamma(b_q) }{ \Gamma(a_1) \cdots \Gamma(a_p) } \sum_{n=0}^\infty \frac{\Gamma( a_1 + A_1 n )\cdots\Gamma( a_p + A_p n )}{\Gamma( b_1 + B_1 n )\cdots\Gamma( b_q + B_q n )} \, \frac {z^n} {n!}$

becomes _pF_q(z) for A_1...p = B_1...q = 1.

The Fox–Wright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50):

${}_p\Psi_q \left[\begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} ; z \right] = H^{1,p}_{p,q+1} \left[ -z \left| \begin{matrix} ( 1-a_1 , A_1 ) & ( 1-a_2 , A_2 ) & \ldots & ( 1-a_p , A_p ) \\ (0,1) & (1- b_1 , B_1 ) & ( 1-b_2 , B_2 ) & \ldots & ( 1-b_q , B_q ) \end{matrix} \right. \right].$

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.