  In this paper we study a generalization of the class of orthogonal
polynomials on the real line. These polynomials satisfy the following relation:
$(J_5 - \lambda J_3) \vec p(\lambda) = 0$, where $J_3$ is a Jacobi matrix and
$J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive
numbers on the second subdiagonal, $\vec p(\lambda) = (p_0(\lambda),
p_1(\lambda), p_2(\lambda),\cdots)^T$, the superscript $T$ means the
transposition, with the initial conditions $p_0(\lambda) = 1$, $p_1(\lambda) =
\alpha \lambda + \beta$, $\alpha > 0$, $\beta\in\mathbb{R}$. Some
orthonormality conditions for the polynomials $\{ p_n(\lambda) \}_{n=0}^\infty$
are obtained. An explicit example of such polynomials is constructed.
