On algebraic identities between solution matrices of generalized hypergeometric equations

The examples of algebraic identities between solution matrices of generalized hypergeometric equations are found in paper. These identities generate all the algebraic identities between components of solutions of hypergeometric equations in some cases.
Generalized hypergeometric functions (see [1–5]) are defined as
$$ {}_l\varphi_{q}(z)={}_l\varphi_{q}(\vec \nu;\vec\lambda;z)= {}_{l+1}F_{q}\left(\left.{1,\nu_1,\dots,\nu_l\atop\lambda_1,\dots,\lambda_q}\right|z\right)= \sum_{n=0}^\infty \frac{(\nu_1)_n\dots (\nu_l)_n}{(\lambda_1)_n \dots(\lambda_{q})_n} z^n, $$
where $0\leqslant l\leqslant q$, $(\nu)_0=1$, $(\nu)_n=\nu(\nu+1) \dots (\nu+n-1)$, $\vec\nu=(\nu_1,\dots,\nu_l)\in {\mathbb C}^l$, $\vec \lambda\in ({\mathbb C}\setminus{\mathbb Z^-})^q$.
The function ${}_l\varphi_{q}(\vec \nu;\vec\lambda;z)$ satisfies the (generalized) hypergeometric differential equation
$$ {L}(\vec \nu;\vec\lambda;z) y =(\lambda_1-1)\dots(\lambda_q-1), $$
where
$$ {L}(\vec \nu;\vec\lambda;z) \equiv \prod_{j=1}^q(\delta+\lambda_j-1)- z\prod_{k=1}^l(\delta+\nu_k), \delta=z\frac{d}{dz}. $$

The Siegel-Shidlovskii method (see [4], [5]) is one of the main methods in the theory of transcendental numbers. It permits to establish the transcendency and the algebraic independence of the values of entire functions of some class, which contains the functions ${}_l\varphi_{q}(\alpha z^{q-l})$, provided that these functions are algebraically independent over ${\mathbb C}(z)$. F. Beukers, W.D. Brownawell and G. Heckman introduced in paper [6] notions of cogredience and contragredience of differential equations, which are important for determination of algebraic dependence and independence of functions (these notions appeared firstly in paper [7] of E. Kolchin really).
This work contains detailed proof and further development of results connected with cogredience and contragredience, that have been published in notes [8], [9]. Some results in [6] have been revised particularly.