Zeros of quasi-orthogonal ultraspherical polynomials

For each fixed value of λλ in the range −3/2<λ<−1/2−3/2<λ<−1/2, we prove interlacing properties for the zeros of polynomials, of consecutive and non-consecutive degree, within the sequence of quasi-orthogonal order 22 ultraspherical polynomials {Cn(λ)}n=0∞. We investigate the manner in which interlacing occurs between the zeros of quasi-orthogonal order 22 ultraspherical polynomials Cn(λ) and their orthogonal counterparts Cm(λ+1) and derive necessary and sufficient conditions for interlacing to occur between the zeros of Cn(λ) and Cm(λ+2),n,m∈N,−3/2<λ<−1/2. In some cases we get interlacing when we include additional factors such as x,1−x2x,1−x2, and occasionally more complicated factors to the polynomials being considered.