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 Ψ q [ ( a 1 , A 1 ) ( a 2 , A 2 ) … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … ( b q , B q ) ; z ] = ∑ n = 0 ∞ Γ ( a 1 + A 1 n ) ⋯ Γ ( a p + A p n ) Γ ( b 1 + B 1 n ) ⋯ Γ ( b q + B q n ) z n n ! . Its normalisation
p Ψ q ∗ [ ( a 1 , A 1 ) ( a 2 , A 2 ) … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … ( b q , B q ) ; z ] = Γ ( b 1 ) ⋯ Γ ( b q ) Γ ( a 1 ) ⋯ Γ ( a p ) ∑ n = 0 ∞ Γ ( a 1 + A 1 n ) ⋯ Γ ( a p + A p n ) Γ ( b 1 + B 1 n ) ⋯ Γ ( b q + B q n ) z n n ! 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 Ψ q [ ( a 1 , A 1 ) ( a 2 , A 2 ) … ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) … ( b q , B q ) ; z ] = H p , q + 1 1 , p [ − z | ( 1 − a 1 , A 1 ) ( 1 − a 2 , A 2 ) … ( 1 − a p , A p ) ( 0 , 1 ) ( 1 − b 1 , B 1 ) ( 1 − b 2 , B 2 ) … ( 1 − b q , B q ) ] . 