The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also multi-indexed Laguerre and Jacobi polynomials, which are new obtained by deforming these classical polynomials. could be considered as a more encompassing characterization of than As start at positive degree ℓD≥1, broken. extra ℓD “lower polynomials,” nece...

We consider a modiication of moment functionals for some classical polynomials of a discrete variable by adding a mass point at x = 0. We obtain the resulting orthogonal polynomials, identify them as hypergeometric functions and derive the second order diierence equation which these polynomials satisfy. The corresponding tridiagonal matrices and associated polynomials were also studied. x1 Intr...

Even though the theory of orthogonal polynomials on the unit circle, also known as the theory of Szegő polynomials, is very extensive, it is less known than the theory of orthogonal polynomials on the real line. One reason for this may be that “beautiful” examples on the theory of Szegő polynomials are scarce. This is in contrast to the wonderful examples of Jacobi, Laguerrer and Hermite polyno...

In this work we give an alterative proof of one of basic properties of zonal polynomials and generalised it for Jack polynomials

We use a generalised Sturmian sequence argument and the discrete orthogonality of the Krawtchouk polynomials for certain parameter values to prove that all the zeros of Meixner polynomials are real and positive for parameter ranges where they are no longer orthogonal. AMS MOS Classification: 33C45, 34C10, 42C05

Following the classical approach of Pólya-Schur theory [14] we initiate in this paper the study of linear operators acting on R[x] and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.

Based on an inversion of the Routh table construction, a unimodular characterization of all Hurwitz polynomials is obtained. In parameter space, the Hurwitz polynomials of degree n correspond to the positive 2-tant. The method is then used to construct classes of stable continuous-time delay-difference equations and delay differential equations of neutral type, by a suitable limiting process.

Equivalent conditions are given for the nonnegativity of the coefficients of both the Chebyshev expansions and inversions of the first n polynomials defined by a certain recursion relation. Consequences include sufficient conditions for the coefficients to be positive, bounds on the derivatives of the polynomials, and rates of uniform convergence for the polynomial expansions of power series.

We decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Gr...

Pólya proved that if a real, homogeneous polynomial is positive on the nonnegative orthant (except at the origin), then it is the quotient of two homogeneous polynomials with no negative coefficients. We generalize this from polynomials to signomials with arbitrary rational exponents; we also show that Pólya’s theorem does not generalize to arbitrary signomials (i.e., with irrational (real) exp...