Asymptotic analysis of a family of polynomials associated with the inverse error function
The authors present a deep study on the behavior of a family of polynomials \((P_{n})_{n\geq0}\) arising when computing higher derivatives of the inverse error function. Indeed, it is equivalent to estimate higher derivatives of the error function at 0 and the corresponding polynomial \(P_{n}(x)\) at \(x=0\) asymptotically in \(n\). The mentioned polynomials satisfy the following recurrence: NEWLINE\[NEWLINE P_0(x)\equiv1,\qquad P_{n+1}(x)=P'_{n}(x)+x(n+1)P_{n}(x),\quad n\geq1. NEWLINE\]NEWLINE The main aim of the work is to obtain different asymptotic expansions for \(n\to\infty\) and (i) \(0<x<\infty\); (ii) \(x=O(n^{-1})\); (iii) \(x=O(\sqrt{\ln(n)})\).NEWLINENEWLINEAfter a singularity analysis for \(P_n(x)\), the WKB method is applied to the previous recurrence satisfied by \((P_{n}(x))_{n\geq0}\) and the fact that \(P_{n}(x)\sim n!x^{n}\), \(x\to\infty\) in order to infer the asymptotics of \(P_n(x)\). Then, the discrete form of the ray method is used to obtain the mentioned uniform asymptotic approximations for \(P_n(x)\).NEWLINENEWLINESeveral plots comparing the estimates and the real values of \(P_n(x)\) in different cases are presented.
scientific article; zbMATH DE number 6066664
scientific article