Saddlepoint expansions in terms of Bell polynomials

Abstract

We express the general term of the asymptotic expansions for

where

in terms of Bell polynomials. Explicit formulas are given when X=A or B and when A<X<B. We assume that ℜ e<texlscub>F (x)</texlscub>is minimized uniquely in [A, B] at x=X, and<texlscub>F(x)−F (X)</texlscub>/|x−X| ^M is analytic at X for some M>0. There is no need to treat the cases M=1, 2, 3, … separately. We also assume that G(x) is analytic at X. These expansions apply when X→∞, as well as for the classic case of fixed [A, B] and F(x)=θ f (x) with θ→∞.

Keywords:

• asymptotic
• Bell polynomials
• compound Poisson gamma
• expansion
• integral
• saddlepoint

Acknowledgements

The authors thank the editor and the referee for careful reading and for their comments which greatly improved the paper.