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Table 2 The effect of the well-known Erdélyi-Kober operators on Meijer's G p , q + 1 1 , p -function

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 )

I 1 , 1 - 1 , 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 )

2 I 1 , 1 0 , 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 )

1 Γ ( α + 1 ) D 1 , 1 - 1 , α

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 )