bibkey: sitnik1993expremainder
authors: S. M. Sitnik
year: 1993
title: Inequalities for the exponential remainder of the Taylor series
doi: null
url: https://arxiv.org/abs/1608.08784
claim: "Problem 11 conjectures positivity of every positive-order forward difference of the consecutive exponential Taylor remainder ratio at each positive argument."
strata_touched:
  - D5/S3/Analytic/ExpRemainderDifferences
license: citation-only
triage: anchor

Exponential remainder ratio differences

Verified locator

Source: https://arxiv.org/abs/1608.08784

Pinned version: https://arxiv.org/abs/1608.08784v2. The original work is the Vladivostok preprint of 1993, titled Неравенства для остатка ряда Тейлора экспоненты; the arXiv version includes a supplement from 2016. The locator is the appendix (Дополнение), additional problem list, Problem 11, PDF pages 27–28. The remainder notation is equation (1); the first-difference inequality is equation (42) in Theorem 8.

Problem 11: source and translation

11. Определим функцию при x > 0. gₙ(x) = Rₙ₋₁(x)/Rₙ(x). Исследовать, при каких k разности этой функции при любом фиксированном x неотрицательны ∆ᵏgₙ(x) > 0? Гипотеза: при любых k. При k = 1 гипотеза сводится к (42), при k = 2 к неравенству задачи 1 на стр. 20.

Translation: For $x>0$, define $g_n(x)=R_{n-1}(x)/R_n(x)$. Investigate for which $k$ the differences of this function, at every fixed $x$, are nonnegative, ${\Delta }^k{g}_n(x)>0$. Conjecture: for all $k$. For $k=1$ the conjecture reduces to (42), and for $k=2$ to the inequality of Problem 1 on page 20.

Mathematical correspondence

Equation (1) defines

$$R_n(x)=e^x-\sum _{j=0}^n\frac{x^j}{j!},x\ge 0.$$

ExpRemainderDifferences.R is precisely this finite-sum expression over the reals. g is $R_{n-1}/R_n$, used at $n\ge 1$ and $x>0$. The forward difference is in the index $n$, with positive unit step:

$${\Delta }^k{g}_n(x)=\sum _{j=0}^k(-1{)}^{k-j}(\genfrac{}{}{0ex}{}{k}{j})g_{n+j}(x).$$

The source calls the differences nonnegative but displays a strict inequality. claim uses the weaker nonnegative assertion, universally over $k\ge 1$, $n\ge 1$ and $x>0$, with Mathlib’s iterated fwdDiff. The negative third difference at $(n,x)=(1,1)$ contradicts both readings. result states exactly the negation of claim.

Equation (42), $R_n^2>R_{n-1}{R}_{n+1}$, gives the case $k=1$. The source identifies $k=2$ with Problem 1; the follow-up reference is K. Mehrez and S. M. Sitnik (2014), https://arxiv.org/abs/1410.6120. The bibliographic scope and unresolved ranges are stated in Problems/sitnik-1993-exponential-remainder-differences.md.