Two-Step Methods and Bi-Orthogonality

We study order and zero-stability of two-step methods of Obrechkoff type for ordinary differential equations. A relation between order and properties of mth degree polynomials orthogonal to <!-- MATH ${x^{{\mu _i}}}$ --> , <!-- MATH $1 \leqslant i \leqslant m$ --> , where <!-- MATH $- 1 < {\mu _1} < {\mu _2} < \cdots < {\mu _m}$ --> <img width="230" height="37" align="MIDDLE" border="0" src="images/img3.gif" alt="$ - 1 < {\mu _1} < {\mu _2} < \cdots < {\mu _m}$">, is established. These polynomials are investigated, focusing on their explicit form, Rodrigues-type formulae and loci of their zeros.