Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userP-finite Recurrences From Generating Functions with Roots of Polynomials
67
0
0.0
(
0
)
Authors
Richard J. Mathar
Ask ChatGPT about the research
No Arabic abstract
We derive the P-finite recurrences for classes of sequences with ordinary generating function containing roots of polynomials. The focus is on establishing the D-finite differential equations such that the familiar steps of reducing their power series expansions apply.
rate research
Read More
Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont-Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials.
It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation. In particular, a numerical observation by Klemev{s} is confirmed.
The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed.
The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-infty,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$. Also we provide a simple analytic proof that for any real $p$ and $r$ with $p>0$, the Fuss-Catalan or Raney numbers $frac{r}{pn+r}binom{pn+r}{n}$, $n=0,1,ldots$ are the moments of a probability distribution on some interval $[0,tau]$ {if and only if} $pge1$ and $pge rge 0$. The same statement holds for the binomial coefficients $binom{pn+r-1}n$, $n=0,1,ldots$.
Various methods can obtain certified estimates for roots of polynomials. Many applications in science and engineering additionally utilize the value of functions evaluated at roots. For example, critical values are obtained by evaluating an objective function at critical points. For analytic evaluation functions, Newtons method naturally applies to yield certified estimates. These estimates no longer apply, however, for Holder continuous functions, which are a generalization of Lipschitz continuous functions where continuous derivatives need not exist. This work develops and analyzes an alternative approach for certified estimates of evaluating locally Holder continuous functions at roots of polynomials. An implementation of the method in Maple demonstrates efficacy and efficiency.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
