Asymptotics of zeros of polynomials arising from rational integrals

Asymptotics of Zeros of Polynomials Arising from Rational Integrals

We prove that the zeros of the polynomials Pm(a) of degree m, defined by Borosh and Moll via Pm(a) = 2m+3/2 π (a + 1) Z ∞ 0 dx (x4 + 2ax2 + 1)m+1 , approach the lemniscate {ζ ∈ C : |ζ − 1| = 1, Rζ < 0}, as m diverges.

Partition Polynomials: Asymptotics and Zeros

Let Fn(x) be the partition polynomial ∑k=1 pk(n)x where pk(n) is the number of partitions of n with k parts. We emphasize the computational experiments using degrees up to 70,000 to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of Fn(x) have two scales of orders n and √ n and in three different regimes inside the unit disk. Consequently, the zeros converge to netw...

On the Zeroes of Two Families of Polynomials Arising from Certain Rational Integrals

Asymptotics of Integrals of Hermite Polynomials

Abstract Integrals involving products of Hermite polynomials with the weight factor exp (−x2) over the interval (−∞,∞) are considered. A result of Azor, Gillis and Victor (SIAM J. Math. Anal. 13 (1982) 879–890] is derived by analytic arguments and extended to higher order products. An asymptotic expansion in the case of a product of four Hermite polynomials Hn(x) as n → ∞ is obtained by a discr...

Higher Order Degenerate Hermite-Bernoulli Polynomials Arising from $p$-Adic Integrals on $mathbb{Z}_p$

Our principal interest in this paper is to study higher order degenerate Hermite-Bernoulli polynomials arising from multivariate $p$-adic invariant integrals on $mathbb{Z}_p$. We give interesting identities and properties of these polynomials that are derived using the generating functions and $p$-adic integral equations. Several familiar and new results are shown to follow as special cases. So...