Polynomials with Odd Orthogonal Multiplicity

Multiplicity of Zeros and Discrete Orthogonal Polynomials

We consider a problem of bounding the maximal possible multiplicity of a zero of some expansions ∑ aiFi(x), at a certain point c, depending on the chosen family {Fi}. The most important example is a polynomial with c = 1. It is shown that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the ze...

Orthogonal Polynomials with Orthogonal Derivatives

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let Λ denote the linear space over R spanned by z, k ∈Z. Define the real inner product (with varying exponential weights) 〈·, ·〉L : Λ ×ΛR→R, ( f , g) 7→ ∫ R f (s)g(s) exp(−N V(s)) ds, N∈N, where the external fieldV satisfies: (i)V is real analytic onR\ {0}; (ii) lim|x|→∞(V(x)/ ln(x2+1))=+∞; and (iii) lim|x|→0(V(x)/ ln(x−2+1))=+∞. Orthogonalisation of the (ordered) base {1, z−1, z, z−2, z2, . . ...

Orthogonal polynomials with discontinuous weights

Abstract. In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, An(z) and Bn(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w0 ...

Orthogonal Polynomials

The theory of orthogonal polynomials plays an important role in many branches of mathematics, such as approximation theory (best approximation, interpolation, quadrature), special functions, continued fractions, differential and integral equations. The notion of orthogonality originated from the theory of continued fractions, but later became an independent (and possibly more important) discipl...