A bivariate approach to realrootedness of special polynomials

In this paper, we exhibit new monotonicity properties of roots of families of orthogonal polynomials $P_n^{(z)}(x)$ depending polynomially on a parameter (Laguerre and Gegenbauer). By establishing that $P_n^{(z)}(x)$ are realrooted in $z$ for $x$ in the support of orthogonality, we show realrootedness in $x$ and interlacing properties of ${\mathrm{\partial }}_z^k{P}_n^{(z)}(x)$ for $k\le n$ and $z\ge 0$, establishing a dual approach to orthogonality.