Related papers: Interlacing and asymptotic properties of Stieltjes…
Polynomial solutions to the generalized Lam\'e equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's in various contexts including the solution of Laplace equations on an ellipsoid.…
A famous result of Stieltjes relates the zeroes of the classical orthogonal polynomials with the configurations of points on the line that minimize a suitable energy. The energy has logarithmic interactions and an external field whose…
We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are…
We investigate the distribution of zeros of the little q-Jacobi polynomials and related q-hypergeometric families. We prove that the zeros of these orthogonal polynomials exhibit strong interlacing properties and obey natural monotonicity…
We investigate the strong asymptotics of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. The solution is given in terms of critical measures (saddle points of…
We present the exact solution for a set of nonlinear algebraic equations $\frac{1}{z_l}= \pi d + \frac{2 d}{n} \sum_{m \neq l} \frac{1}{z_l-z_m}$. These were encountered by us in a recent study of the low energy spectrum of the Heisenberg…
We establish a hierarchy of weighted majorization relations for the singularities of generalized Lam\'e equations and the zeros of their Van Vleck and Heine-Stieltjes polynomials as well as for multiparameter spectral polynomials of higher…
Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death,…
We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra…
The generating function of Stieltjes-Carlitz polynomials is a solution of Heun's differential equation and using this relation Carlitz was the first to get exact closed forms for some Heun functions. Similarly the associated…
In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.
The theory of polynomials orthogonal with respect to one inner product is classical. We discuss the extension of this theory to multiple inner products. Examples include the Lam\'e and Heine-Stieltjes polynomials.
The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of…
We investigate the interlacing of zeros of polynomials of different degrees within the sequences of $q$-Laguerre polynomials $\left\{\tilde{L}_n^{(\delta)}(z;q)\right\}_{n=0}^{\infty}$ characterized by $\delta\in(-2,-1).$ The interlacing of…
It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main…
We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) P\'olya frequency sequences are infinitely log-concave. We introduce the concept of $q$-Stieltjes moment sequences of…
New polynomials associated with a special Lipkin-Meshkov-Glick (LMG) model corresponding to the standard two-site Bose-Hubbard model are derived based on the Stieltjes correspondence. It is shown that there is a one-to-one correspondence…
We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly…
We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials p_n(x) satisfying a difference equation of the form B(x)p_n(x+\delta)-C(x,n)p_n(x)+D(x)p_n(x-\delta)=0. We calculate the asymptotic distribution of…
We give the asymptotic behavior of the zeros of orthogonal polynomials, after appropriate scaling, for which the orthogonality measure is supported on the $q$-lattice $\{q^k, k=0,1,2,3,\ldots\}$, where $0 < q < 1$. The asymptotic…