Zeros of linear combinations of Laguerre polynomials from different sequences

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely $R_n=L_n^{\alpha}+aL_{n}^{\alpha'}$ and $S_n=L_n^{\alpha}+bL_{n-1}^{\alpha'}$. Proofs and numerical counterexamples are given in situations where the zeros of $R_n$, and $S_n$, respectively, interlace (or do not in general) with the zeros of $L_k^{\alpha}$, $L_k^{\alpha'}$, $k=n$ or $n-1$. The results we prove hold for continuous, as well as integral, shifts of the parameter $\alpha$

A Lanczos Method for Approximating Composite Functions

We seek to approximate a composite function h(x) = g(f(x)) with a global polynomial. The standard approach chooses points x in the domain of f and computes h(x) at each point, which requires an evaluation of f and an evaluation of g. We present a Lanczos-based procedure that implicitly approximates g with a polynomial of f. By constructing a quadrature rule for the density function of f, we can approximate h(x) using many fewer evaluations of g. The savings is particularly dramatic when g is much more expensive than f or the dimension of x is large. We demonstrate this procedure with two numerical examples: (i) an exponential function composed with a rational function and (ii) a Navier-Stokes model of fluid flow with a scalar input parameter that depends on multiple physical quantities

Onsager's algebra and partially orthogonal polynomials

The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager's algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the Z_3-case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function.Comment: 8 pages, no figure

A quantum exactly solvable non-linear oscillator related with the isotonic oscillator

A nonpolynomial one-dimensional quantum potential representing an oscillator, that can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends of a parameter $a$, is considered and then a particular case is studied with great detail. It is proven that it is Schr\"odinger solvable and then the wave functions $\Psi_n$ and the energies $E_n$ of the bound states are explicitly obtained. Finally it is proven that the solutions determine a family of orthogonal polynomials ${\cal P}_n(x)$ related with the Hermite polynomials and such that: (i) Every ${\cal P}_n$ is a linear combination of three Hermite polynomials, and (ii) They are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.Comment: 11 pages, 11 figure