Recurrence Relation and Differential Equation for a Class of Orthogonal Polynomials

Given real number \(s>-1/2\) and the second degree monic Chebyshev polynomial of the first kind \(\widehat{T}_{2}(x)\), we consider the polynomial system \(\{p_{k}^{2,s}\}\) “induced” by the modified measure \({\,\mathrm {d}}\sigma ^{2,s}(x)=|\widehat{T}_{2}(x)|^{2s}{\,\mathrm {d}}\sigma (x)\), where \({\,\mathrm {d}}\sigma (x)=1/\sqrt{1-x^2}{\,\mathrm {d}}x\) is the Chebyshev measure of the first kind. We determine the coefficients of the three-term recurrence relation for the polynomials \(p_{k}^{2,s}(x)\) in an analytic form and derive a differential equality, as well as the differential equation for these orthogonal polynomials. Assuming a logarithmic potential, we also give an electrostatic interpretation of the zeros of \(p_{4\nu }^{2,s}(x) (\nu \in {\mathbb N})\).