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Effective Borel-resummation by factorial series. (Sommation effective d’une somme de Borel par séries de factorielles.) (French. English summary) Zbl 1129.30023
In der Theorie der Borelschen Summation läuft die Summation einer Potenzreihe \(\varphi(z)=\sum^\infty_{n=0}a_nz^n\) darauf hinaus statt \(\varphi\) die Reihe \(f(s)=\sum^\infty_{n=0}\frac{a_n} {s^{n+1}}\) zu betrachten und diese als Laplace-Transformierte von \(F (t)=\sum^\infty_{n=0}a_n\frac{t^n}{n!}\) aufzufassen. Unter gewissen Konvergenzbedingungen ergibt sich dann die Integraldarstellung \(f(s)= \frac 1s\int^\infty_0e^{-x}F(\frac xs)\,dx\). Hieraus folgt durch eine Variablensubstitution in einem bestimmten Bereich der komplexen Zahlenebene die Repräsentation von \(f\) durch eine Fakultätenreihe der Form \(f(s)=a_0+ \sum^\infty_{n=0}\frac{n!b_n}{s(s+1)\dots (s+n)}=h(s)\).
Die Autoren geben in Abhängigkeit von \(N\) und vom Konvergenzbereich konkrete Abschätzungen für das Restglied \(R_N\) der Faktultätenreihe \(h(s)\) an, wenn \(h(s)\) nur bis zur Stelle \(n=N\) betrachtet wird. Die ganzen Abhandlungen werden erweitert auf die Borelsche Summation von Potenzreihen mit gebrochenen Exponenten, ferner werden spezielle Beispiele angegeben.

MSC:
30E15 Asymptotic representations in the complex plane
40G99 Special methods of summability
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
40G10 Abel, Borel and power series methods
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