bibkey: zhu2020completelymonotonicdegrees
authors: L. Zhu
year: 2020
title: "Completely monotonic integer degrees for a class of special functions"
doi: 10.3934/math.2020224
url: https://www.aimspress.com/article/doi/10.3934/math.2020224
claim: "Conjecture 6.4 asserts positivity at derivative order m+2n-2 and failure of positivity everywhere at order m+2n-1 for the signed Bernoulli remainder, for integers m >= 1 and n >= 2."
strata_touched:
  - D5/S3/Analytic/CothBernoulliDegree/Defs
  - D5/S3/Analytic/CothBernoulliDegree/Result
license: citation-only
triage: anchor

Completely monotonic integer degrees of a hyperbolic cotangent remainder

Verified locator

L. Zhu, Completely monotonic integer degrees for a class of special functions, AIMS Mathematics 5(4) (2020), 3456–3471.

DOI: 10.3934/math.2020224

Source: https://www.aimspress.com/article/doi/10.3934/math.2020224

Conjecture 6.4, equations (6.12) and (6.13), uses the remainder defined in equation (2.2).

Mathematical correspondence

Equation (2.2) defines

$$p_n(t)=\frac{\coth t}{t}-\sum _{k=0}^n\frac{2^{2k}{B}_{2k}}{(2k)!}{t}^{2k-2}.$$

For integers $m\ge 1$, $n\ge 2$, put $M=m+2n$ and $P(t)=(-1{)}^n{t}^m{p}_n(t)$. Conjecture 6.4 states that $P^{(M-2)}(t)>0$ for every $t>0$, while $P^{(M-1)}(t)>0$ does not hold for every $t>0$. The two clauses are equations (6.12) and (6.13).

Defs.p, Defs.P and Defs.claim in D5/S3/Analytic/CothBernoulliDegree/Defs retain these definitions and quantifiers. Integer powers are used in the finite remainder sum, including $t^{-2}$ at $k=0$. Only even Bernoulli indices occur, so the convention for $B_1$ does not affect the remainder. D5/S3/Analytic/CothBernoulliDegree/Result.result proves both clauses.

The bounded literature scope and its limitations are stated in Problems/zhu-2020-coth-bernoulli-remainder-derivative-signs.md.