From: Fractional differintegral transformations of univalent Meijer's G-functions
Operators
Transformation
I 1 , 1 - 1 , 1 (Biernacki)
G 1 , 1 1 , 1 → G 2 , 2 1 , 2 [ 0 , 0 1 − a 1 , 1 | − z ] ( b 1 = 1 , a 2 = 0 , b 1 > a 2 ; p = 1 , q = 0 )
2 I 1 , 1 0 , 1 (Libera)
G 1 , 1 1 , 1 → 2 G 2 , 2 1 , 2 [ 0 , − 1 1 − a 1 , 0 | − z ] ( b 1 = 2 , a 2 = 1 , b 1 > a 2 ; p = 1 , q = 0 )
1 Γ ( α + 1 ) D 1 , 1 - 1 , α (Ruscheweyh)
G 1 , 1 1 , 1 → 1 Γ ( α + 1 ) G 2 , 2 1 , 2 [ 0 , 1 1 − a 1 , 1 − α | − z ] ( b 1 = 0 , a 2 = α , ; p = 2 , q = 1 )
I 1 , 1 - 1 , 1
G 1 , 2 1 , 1 → G 2 , 3 1 , 2 [ 0 , 1 − b 1 , 0 1 − a 1 , 1 | − z ] ( b 2 = 1 , a 2 = 0 , b 2 > a 2 ; p = q = 1 )
2 I 1 , 1 0 , 1
G 1 , 2 1 , 1 → 2 G 2 , 3 1 , 2 [ 0 , 1 − b 1 , − 1 1 − a 1 , 0 | − z ] ( b 2 = 2 , a 2 = 1 , b 2 > a 2 ; p = q = 1 )
1 Γ ( α + 1 ) D 1 , 1 - 1 , α
G 1 , 2 1 , 1 → 1 Γ ( α + 1 ) G 2 , 3 1 , 2 [ 0 , 1 − b 1 , 1 1 − a 1 , 1 − α | − z ] ( b 2 = 0 , a 2 = α ; p = q = 1 )
G 0 , 2 1 , 0 → G 1 , 3 1 , 1 [ 0 , 1 − b 1 , 0 1 | − z ] ( b 2 = 1 , a 1 = 0 , b 2 > a 1 ; p = 0 , q = 1 )
G 0 , 2 1 , 0 → 2 G 1 , 3 1 , 1 [ 0 , 1 − b 1 , − 1 0 | − z ] ( b 2 − 2 , a 1 = 1 , b 2 > a 1 ; p = 0 , q = 1 )
G 0 , 2 1 , 0 → 1 Γ ( α + 1 ) G 1 , 3 1 , 1 [ 0 , 1 − b 1 , 1 1 − α | − z ] ( b 2 = 0 , a 1 = α ; p = 0 , q = 1 )