Zeros of a binomial combination of Chebyshev polynomials

Abstract

For $0<\alpha <1$, we study the zeros of the sequence of polynomials ${\left\{P_m(z)\right\}}_{m=0}^{\infty }$ generated by the reciprocal of $(1-t{)}^{\alpha }(1-2zt+t^2)$, expanded as a power series in $t$. Equivalently, this sequence is obtained from a linear combination of Chebyshev polynomials whose coefficients have a binomial form. We show that the number of zeros of $P_m(z)$ outside the interval $(-1,1)$ is bounded by a constant independent of $m$.