A criterion for the algebraic independence of values of a class of hypergeometric $E$-functions

A criterion for the irreducibility of the equation
$$ \biggl\{\biggl(z\frac\partial{dz}+\lambda_1\biggr)\dotsb\biggl(z\frac\partial{dz}+\lambda_{t+l}\biggr)-z^t\biggl(z\frac\partial{dz}+\nu_1\biggr)\dotsb \biggl(z\frac\partial{dz}+\nu_l\biggr)\biggr\}(y)=0 $$
is established in the case when $l\geqslant 0$ and $t$ is odd. Arithmetical applications of this result are obtained.