Permanence de relations de récurrence dans certains développements asymptotiques. (Permanence of recurrence relations in certain asymptotic expansions). (French) Zbl 0655.10040

Robin, Guy

Publ. Inst. Math., Nouv. Sér. 43(57), 17-25 (1988).

J.-P. Massias, J.-L. Nicolas and G. Robin [Acta Arith. 50, No.3, 221-242 (1988; Zbl 0588.10049)] approximated certain arithmetical functions by \((Li^{-1}(x))^{1/2}\) resp. \(Li((Li^{- 1}(x))^{1/2}).\) Their asymptotic expansions motivated the following investigations.
Let us consider a function f having an asymptotic expansion of the form \[ f(x)=e\quad x x^{-\alpha}(D_ N(x^{-1})+o(x^{-N}))\quad for\quad x\to \infty, \] where \(\alpha\in R\) *, \(D_ N\in R[x]\), deg \(D_ N\leq N\), \(D_ N(0)\neq 0\) and suppose that f has an inverse function g. Theorem 1 of the paper under review gives recurrence relations of type \(P'_{n+1}=P'_ n-nP_ n\) for the polynomials \(P_ n\) occurring in the asymptotic expansion of g. Theorem 2 gives an analogous result for \[ F(x)=x(\log x)^{-\alpha} (D_ N((\log x)^{- 1})+o((\log x)^{-N})). \] (Propositions 1 and 2 yield the asymptotic expansions of log f, \(f^{\beta}\) and \(h\circ f\) for certain functions f and h.) Theorems 1 and 2 are supplemented by programs calculating the polynomials in question.

Reviewer: M.Szalay

MSC:

11N37	Asymptotic results on arithmetic functions
41-04	Software, source code, etc. for problems pertaining to approximations and expansions
41A10	Approximation by polynomials
41A60	Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Keywords:

asymptotic expansions of inverse functions; recurrence relations for polynomials; MACSYMA; arithmetic functions

Citations:

Zbl 0588.10049

Software:

MACSYMA