is not considered to be summed up. In order to uniquely define this transverse component (index), we introduce a light-like vector n µ h ∝ ( P 0 $_{h}$, − ⃗ P$_{h}$ ) with normalization that n$_{h}$ · P$_{h}$ = 1 $^{2}$. Here and in the following, we use subscript “ T ” to distinguish the transverse direction perpendicular to P$_{h}$ from “ ⊥ ” in Eq. (2) for transverse direction perpendicular to P$_{A}$ and P$_{B}$ . Therefore, the transverse component for any vector p µ can be defined as p µ T = p µ − p · n$_{h}$P µ h − p · P$_{h}$n µ $_{h}$. From this definition, we can immediately see that p$_{T}$ is space-like, and perpendicular to ⃗ P$_{h}$ : ⃗ p$_{T}$ · ⃗ P$_{h}$ = 0. As discussed in Ref. [23], the above twist-three fragmentation function belongs to more general two-variable dependent twist-three fragmentation function defined as, where k + i = P + h /z$_{i}$ and k + g = k + 1 − k + 2 . Similarly, we can define a F -type fragmentation function ˆ H$_{F}$ ( z$_{1}$, z$_{2}$ ) by replacing D α T with F + α in Eq. (5). D -type and F -type functions are related to each other, From this equation, we can see that ˆ H$_{D}$ is more singular than ˆ H$_{F}$ at z$_{1}$ = z$_{2}$ . Moreover, recent study has shown that ˆ H$_{F}$ ( z$_{1}$, z$_{2}$ ) vanishes when z$_{1}$ = z$_{2}$ [31]. Therefore, it is more convenient to express the cross section in terms of ˆ H ( z ) and ˆ H$_{F}$ ( z$_{1}$, z$_{2}$ ), which we will follow. In this paper, we are interested in obtaining the phenomenological important contribu- tions in hadron production in the forward region of the polarized beam. Similar to the twist-three distribution function contributions, we find that the twist-three fragmentation function contributions also contain a derivative term which certainly will dominate this part of contribution in the forward rapidity region. In addition, the derivative contribution from the twist-three fragmentation function is associated with ˆ H ( z ) defined in Eq. (3), whereas the contribution associated with ˆ H$_{F}$ ( z$_{1}$, z$_{2}$ ) does not contain any derivative contributions. In 5 2 In the low transverse momentum semi-inclusive hadron production in DIS process, n h will be proportional to the incoming nucleon momentum as we used in Ref. [23]. In current study, there is no natural available momentum for this choice. We can also perform the calculations without choosing the vector n h , but keeping all transverse component in Eq. (3) perpendicular to the hadron momentum. This will lead to the same results. ˆ H$_{D}$ ( z$_{1}$, z$_{2}$ ) = P V ( 1 1 z 1 − 1 z 2 ) ˆ H$_{F}$ ( z$_{1}$, z$_{2}$ ) + δ ( 1 z$_{1}$ − 1 z$_{2}$ ) z$_{2}$ z$_{1}$ ˆ H ( z$_{1}$ ) . (6) ˆ H$_{D}$ ( z$_{1}$, z$_{2}$ ) = z$_{1}$z$_{2}$ 2 ∫ dξ $^{−}$dζ − (2 π ) 2 e ik + 2 ξ $^{−}$e ik + g ζ − 1 2 { Tr σ α $^{+}$〈 0 | iD α $_{T}$( ζ $^{−}$) ψ ( ξ $^{−}$) | P$_{h}$X 〉 × 〈 P$_{h}$X | ¯ ψ (0) | 0 〉 + h.c. } , (5)