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

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.)

Zbl 0588.10049

MACSYMA
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