Monotonicity of the zeros of orthogonal polynomials through related measures (Q557022)

In this quite interesting paper, the authors use known facts about orthogonal polynomials where the associated measures on the real line are related by a linear polynomial factor (in numerator/denominator, of constant sign ), specifically the three term recurrence relation containing one `old' polynomial and two `new' ones, to derive results on the monotonicity of the zeros. Let \(d\phi\) be a determinate measure on \([a,b]\;(-\infty<a<b\leq \infty)\) (for \(b<\infty\) the measure is always determinate) and let \(N,\kappa\) be real numbers with \(0\leq N<\infty,\;\kappa<a\) (or exceptionally \(\kappa\leq a\)). Apart from the monic orthogonal polynomials \(P_n^{(\phi)}\) associated with \(d\phi\), consider the monic orthogonal polynomials \(P_n^{(\phi,\kappa,N)}\) with associated measure given by \[ \int p(x)d\phi^{\kappa,N)}(x)={1\over N+1}\,\left\{Np(\kappa)+{{\mathcal M}^{(\phi,\kappa)}[p] \over {\mathcal M}^{(\phi,\kappa)}[1]}\right\}, \] where \(p\) is an arbitrary polynomial and \[ {\mathcal M}^{(\phi,\kappa)}[p]=\int_a^b\,p(x)(x-\kappa)^{-1}d\phi(x). \] The value \(\kappa=a\) is allowed, only if \({\mathcal M}^{(\phi,\kappa)}[1]\) is convergent. The main result of the paper is then a theorem on the monotonicity behavior of the zeros of the monic orthogonal polynomials: Let \(x_{n,r}^{(\phi,\kappa,N)}\) and \(x_{n,r}^{(\phi)},\;1\leq r\leq n\), be the respective zeros of \(P_n^{(\phi,\kappa,N)}\) and \(P_n^{(\phi)}\), arranged in \textit{decreasing order}. Then \textbf{1.} \(x_{n,r}^{(\phi)}<x_{n,r}^{(\phi,\kappa,N)},\;1\leq r\leq n\) and \(\kappa < x_{n,n}^{(\phi,\kappa,N)}<x_{n,n}^{(\phi)}\). \textbf{2.} \(x_{n,r}^{(\phi,\kappa,N)}\) is a decreasing function of \(N\). \textbf{3.} If \(N=0\), then \(x_{n,r}^{(\phi,\kappa,N)}\) is a decreasing function of \(\kappa\). \textbf{4.} If \(N>0\), then \(x_{n,r}^{(\phi,\kappa,N)}\) is an increasing function of \(\kappa\), provided that \(\kappa\) varies in the range \((-\infty,\check{\kappa}^{(\phi,N)}]\), where \[ \check{\kappa}^{(\phi,N)}=a-{\sqrt{[N(\beta_2^{(\phi)}-a)-(\beta_1^{(\phi)}-a)]^2+4Nv_1^{(\phi)}(a)} - [N(\beta_2^{(\phi)}-a)-(\beta_1^{(\phi)}-a)]\over 2N}. \] Here \(v_1^{(\phi)}=(\beta_1^{(\phi)}-a)(\beta_2^{(\phi)}-a)-\alpha_2^{(\phi)}>0\) and \(\alpha_n^{(\phi)},\,\beta_n^{(\phi)}\) are the coefficients from the three term recurrence relation for the \(P_n^{(\phi)}\). This theorem is then applied to the cases of the Jacobi, Laguerre and Charlier polynomials. The functionals \({\mathcal M}\) are respectively given by \[ {\mathcal M}_J^{(\alpha,\beta,\kappa)}[p]=\int_{-1}^1\,p(x)\,{(1-x)^{\alpha}(1+x)^{\beta}\over x-\kappa}\,dx, \] \[ {\mathcal M}_L^{(\alpha,\kappa)}[p]=\int_0^{\infty}\,p(x)\,{x^{\alpha}e^{-x}\over x+\kappa}\,dx, \] \[ {\mathcal M}_C^{(\alpha,\kappa)}[p]=\int_0^{\infty}\,p(x)\,d\phi_C^{(\alpha)}(x) = \sum_{r=0}^{\infty}\,p(r)\,{1\over r+\kappa}\,{e^{-\alpha}\alpha^r\over r!}. \]