On algebraic identities between solution matrices of generalized hypergeometric equations

Abstract

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 differen\-tial equation
$$
{L}(\vec \nu;\vec\lambda;z)\;y =(\lambda_1-1)\dots(\lambda_q-1),
$$
where
$$
{L}(\vec \nu;\vec\lambda;z)
\equiv \prod_{j=1}^q(\delta+\lambda_j-1)-
z\prod_{k=1}^l(\delta+\nu_k), \quad \delta=z\frac{d}{dz}.
$$

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.