on pexider type of hilbert c$^* $-module higher derivations

In this paper, we introduce the concept of pexider hilbert c$ ^* $-module higher $ {a_n,b_n,d_n} $-derivations. specifically, we focus on a hilbert c$ ^* $-module $ mathcal{m} $ and provide a comprehensive characterization of these pexider hilbert c$ ^* $-module higher $ {a_n,b_n,d_n} $-derivations $ {phi_n}_{n=0}^infty $ on $ mathcal{m} $ in relation to pexider hilbert c$ ^* $-module $ {alpha_n, beta_n, delta_n} $-derivations $ {varphi_n }_{n=1}^infty$ on $ mathcal{m} $. we demonstrate that for every pexider hilbert c$ ^* $-module higher $ {a_n,b_n,d_n} $-derivation $ {phi_n}_{n=0}^infty $ on $ mathcal{m} $, there exists a unique sequence of pexider hilbert c$ ^* $-module $ {alpha_n, beta_n, delta_n} $-derivations $ {varphi_n }_{n=1}^infty$ on $ mathcal{m} $ such that$$left{begin{array}{lr}varphi_n=sum_{k=1}^nbig(sum_{sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1phi_{r_1}phi_{r_2}ldots phi_{r_k}big),alpha_n=sum_{k=1}^nbig(sum_{sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1a_{r_1}a_{r_2}ldots a_{r_k}big),beta_n=sum_{k=1}^nbig(sum_{sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1b_{r_1}b_{r_2}ldots b_{r_k}big),delta_n=sum_{k=1}^nbig(sum_{sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1d_{r_1}d_{r_2}ldots d_{r_k}big),end{array}right.$$for all positive integers $ n$, where the inner summation is taken over all positive integers $r_j$ with $sum_{j=1}^k r_j=n$.