Asymptotic analysis of a family of polynomials associated with the inverse error function

Abstract

We analyze the sequence of polynomials defined by the differentialdifference equation P n+1 (x) = P ′ n (x) + x(n + 1)P n (x) asymptotically as n → ∞. The polynomials P n (x) arise in the computation of higher derivatives of the inverse error function inverf(x). We use singularity analysis and discrete versions of the WKB and ray methods and give numerical results showing the accuracy of our formulas.MSC-Class: 33B20 (Primary) 34E20, 33E30 (Secondary).

Zero distribution of polynomials satisfying a differential-difference equation

“…In Figure 2 , we plot the zero counting measure ψ 100 (z) defined in (44) and the measure ψ (t) defined in (66). In [8], we analyzed the polynomials P n (x) asymptotically and, among others, we considered the limit n → ∞, with x = y/n and y = O(1). We obtained the asymptotic approximation P n (x) ∼ n n e −n 2n ln(n) 2 π n exp 2 π y + (−1) n exp − 2 π y , and therefore Q n (x) ∼ i n 1 π ln(n) 2 π n exp − 2 π nxi + (−1) n exp 2 π nxi ,…”

“…In [8], we analyzed the polynomials P n (x) asymptotically and, among others, we considered the limit n → ∞, with x = y/n and y = O(1). We obtained the asymptotic approximation where we have used Stirling's formula [17, 5.11.7] n!…”

Analytic Error Function and Numeric Inverse Obtained by Geometric Means

“…Known approximations are dating back to the late 1960s and early 1970s [9,10]) and reach up to semianalytical approximations by asymptotic expansion (cf., e.g., Refs. [11][12][13][14][15]Soranzo]. Using the same geometric considerations, in Sec.…”

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

“…In a series of papers [7,8,10,13,14,16], we studied polynomial solutions of differentialdifference equations of the form…”