Monotonicity of zeros of polynomials orthogonal with respect to an even weight function

The monotonicity properties of all the zeros with respect to a parameter of orthogonal polynomials associated with an even weight function are studied. The results we obtain extend the work of A. Markoff. The monotonicity of the zeros of Gegenbauer, Freud-type and symmetric Meixner–Pollaczek orthogonal polynomials as well as Al-Salam–Chihara q-orthogonal polynomials are investigated. For the Meixner–Pollaczek polynomials, a special case of a conjecture by Jordaan and Toókos which concerns the interlacing of their zeros between two different sequences of Meixner–Pollaczek polynomials is proved.