Expansion in Eigenfunctions of Quadratic Strongly Irregular Pencils of Differential Operators of the Second Order

We consider a quadratic strongly irregular pencil of $2$-d order ordinary differential operators with constant coefficients and positive roots of the characteristic equation. Both the amounts of double expansions in a series in the derivative chains of such pencils and necessary and sufficient conditions for convergence of these expansions to the decomposed vector-valued function are found.