On a system of functional equations (Q5902818)

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 \(U(t)=\sum^{\infty}_{k=0}u_ kt^{4k+2}\).