bibkey: baowangqiu2022ellipticparameter
authors: Q. Bao, M.-K. Wang and S.-L. Qiu
year: 2022
title: "Monotonicity properties of the generalized elliptic integrals with respect to the parameter"
doi: 10.7153/mia-2022-25-64
url: https://arxiv.org/abs/2112.13993
claim: "Open problem 3.5(2) asks for the supremum of the remainder parameter exponent lambda10 for every positive truncation order."
strata_touched:
  - D5/S3/Analytic/EllipticTailExponent/Defs
  - D5/S3/Analytic/EllipticTailExponent/Result
license: citation-only
triage: anchor

Elliptic remainder parameter exponent

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DOI: 10.7153/mia-2022-25-64

Source: https://arxiv.org/abs/2112.13993

Pinned version: https://arxiv.org/abs/2112.13993v1, equations (3.2) and (3.6), Lemma 3.2(4), Remark 3.4 and Open problem 3.5(2). The journal version is Mathematical Inequalities & Applications 25(4) (2022), 1021–1045.

Mathematical correspondence

With $(a{)}_k={\prod }_{j=0}^{k-1}(a+j)$, equation (3.2) at $c=1$ gives $g_{6,k}(a)=a(a{)}_k(1-a{)}_k$. Put

$$S_n(a)=\sum _{k=0}^n\frac{g_{6,k}(a)}{(k+1)(k!{)}^2}.$$

Equation (3.6) defines

$${\lambda }_{10}(a)=\frac{a}{1-a}\frac{\sin (\pi a)+\pi (1-a)\cos (\pi a)-\pi (1-a{)}^2{S}_n^{\prime }(a)}{\sin (\pi a)-\pi (1-a)S_n(a)}.$$

Defs.g6, Defs.S and Defs.lambda10 in D5/S3/Analytic/EllipticTailExponent/Defs encode these real functions, including the ordinary real derivative of the finite polynomial. Defs.claim asserts that two is the least upper bound of the exponent values on $(0,1/2]$ for every natural $n$, including zero. D5/S3/Analytic/EllipticTailExponent/Result.result proves this claim. Lemma 3.2(4) settles $n=0$; Open problem 3.5(2) asks for the general value.

The literature scope and the distinction between the formal result and its further consequences are stated in Problems/bao-wang-qiu-2022-elliptic-tail-exponent.md.