Algebraic relations between the hypergeometric \(E\)-function and its derivatives (Q1810138)

The second author continues his study of algebraic independence of the values of hypergeometric \(E\)-functions, defined by \[ \varphi(z)= \varphi^-_\lambda(z)= \sum^\infty_{n=0} {(z/t)^{tn} \over(1+ \lambda_1)_n \cdots(1+ \lambda_t)_n}, \] with \(t\) being even and \(\lambda_i\in\mathbb{Q}\) and \(-\lambda_i\notin \mathbb{N}\). In the present paper the authors provide a complete structure of algebraic relations over \(\mathbb{C}(z)\) of these functions provided they are algebraic dependent over \(\mathbb{C}(z)\). The results obtained are useful in obtaining certain lower bounds.