On the linear independence of the values of \(E\)-functions (Q1382601)

Let \[ f_1(z),\dots,f_m(z)\tag{1} \] be \(E\)-functions that are solutions of the system of homogeneous differential equations \[ y'_k = \sum\limits_{k=1}^m Q_{k,i}y_i,\quad k = 1,\dots,m,\quad Q_{k,i}\in\mathbb{C}(z),\tag{2} \] and let \(\mathcal L\) be a linear space generated by the numbers \[ f_1(\xi),\dots,f_m(\xi)\tag{3} \] over the field \(K\). The main result of the paper is as follows. If the \(E\)-functions (1) are not connected by a homogeneous algebraic equation over \(\mathbb{C}(z)\) in powers of \(h\) (\([h/2]\), if \(\theta\notin\mathbb{R}\)), then the inequality \(\rho = \dim\mathcal L\geq l+1\) is satisfied, where \(l\) is the homogeneously transcendental degree of the functions (1) over \(\mathbb{C}(z)\), \(1\leq l < m\).