Motivated by the theory of generalized Appel sequences when looking for the square roots of the Laguerre sequence, the authors study the system of functional equations (1) \(H(H(t))=t/(t-1)\); (2) \(G(t)G(H(t))=(1- t)^{-\alpha -1}\), \(\alpha >-1\), in the class of formal power series (3) \(H(t)=\sum^{\infty}_{k=1}h_ kt^ k\), \(G(t)=\sum^{\infty}_{k=0}g_ kt^ k\) with complex coefficients. The following theorem is proved: The formal power series (3) satisfy the system of equations (1), (2), if and only if \(H(t)=\phi^{-1}(\eta \phi (t))\), \(G(t)=((1-t)/(1- H(t)))^{(-\alpha -1)/2}\cdot G^*(t)\) where: \(\eta =+i\) or \(\eta =-i\); \(\phi (t)=B(t)-B(t/(t-1))\), B arbitrary formal power series with B'(0)\(\neq 0\); either \(G^*(t)=\epsilon \frac{L(t)L(H^ 2(t))}{L(H(t))L(H^ 3(t))}\) with \(\epsilon =+1\) or \(\epsilon =-1\) and L arbitrary formal power series with L(0)\(\neq 0\), or \(G^*(t)=\sigma \exp U(\phi (t))\) where \(\sigma =+1\) or \(\sigma =-1\) and U is an arbitrary formal power series of the form \(U(t)=\sum^{\infty}_{k=0}u_ kt^{4k+2}\).