Thus These formal series can be summed exactly as for the case of singlet electric and magnetic quark GPDs. The form of the resulting expression actually differs only by a factor 1 $_{3}$. The expressions for gluon electric and magnetic elementary amplitudes read We introduce electric and magnetic gluon GPD quint-essence functions: The imaginary parts of gluon electric and magnetic elementary amplitudes then read: 16 A g ( E $^{)}$( ξ, t ) = 4 3 ∞ ∑ n =1 odd n +1 ∑ l =0 even B g ( E ) n l P$_{l}$ ( 1 ξ ) ; A g ( M $^{)}$( ξ, t ) = 4 3 ∞ ∑ n =1 odd n +1 ∑ l =0 even B g ( M ) n l 1 ξ $_{P}$′ l ( 1 ξ ) . (43) A g ( E $^{)}$( ξ, t ) = 2 3 ∫ 1 0 dx x ∞ ∑ ν =0 x 2 $^{ν}$G ( E ) 2 ν ( x, t )   1 √ 1 − 2 x ξ + x 2 + 1 √ 1 + 2 x ξ + x 2 − 2 δ$_{ν}$$_{0}$   ; (44) A g ( M $^{)}$( ξ, t ) = ( − ξ ∂ ∂ξ ) 2 3 ∫ 1 0 dx x ∞ ∑ ν =0 x 2 $^{ν}$G g ( M ) 2 ν ( x, t )   1 √ 1 − 2 x ξ + x 2 + 1 √ 1 + 2 x ξ + x 2 − 2 δ$_{ν}$$_{0}$   . (45) N g ( E $^{)}$( x, t ) = ∞ ∑ ν =0 x 2 ν G ( E ) 2 ν ( x, t ) ; N g ( M $^{)}$( x, t ) = ∞ ν =0 x 2 ν G ( M ) 2 ν ( x, t ) . (46) Im A g ( E $^{)}$( ξ, t ) = πH g ( E $^{)}$( ξ, ξ, t ) ξ = 2 3 ∫ 1 1$_{−}$ √ 1$_{−}$ ξ 2 ξ dx x N g ( E $^{)}$( x, t ) [ 1 √ 2 x ξ − x 2 − $_{1}$] ; (47) Im A g ( M $^{)}$( ξ, t ) = πH g ( M $^{)}$( ξ, ξ, t ) ξ = − 2 3 ∫ 1 1$_{−}$ √ 1$_{−}$ ξ 2 ξ dx { ∂ ∂x N g ( M $^{)}$( x, t ) 1 − ξx } [ 1 √ 2 x ξ − x 2 − $_{1}$] ; (48)