Zeros of Taylor polynomials formed by a three-term recurrence

From a sequence ${\left\{a_n\right\}}_{n=0}^{\mathrm{\infty }}$ of real numbers satisfying a three-term recurrence, we form a sequence of polynomials ${\{P_m(z)\}}_{m=0}^{\mathrm{\infty }}$ whose coefficients are numbers in this sequence. We showed that under explicit conditions, the zeros of $P_m(z)$, $m\gg 1$, lie on one side of the circle whose radius is given by modulus of a zero of the denominator of the generating function of ${\left\{a_n\right\}}_{n=0}^{\mathrm{\infty }}$.