bibkey: boudabsasimon2021kilbassaigo
authors: L. Boudabsa and T. Simon
year: 2021
title: "Some properties of the Kilbas–Saigo function"
doi: 10.3390/math9030217
url: https://arxiv.org/abs/2012.05666
claim: "Conjecture 4.5 gives two-sided hyperbolic bounds on the negative half-line for 0 < alpha <= 1, m > 0 and l > m - 1/alpha."
strata_touched:
  - D5/S3/Analytic/KilbasSaigo/Defs
  - D5/S3/Analytic/KilbasSaigo/Result
license: citation-only
triage: anchor

Kilbas–Saigo hyperbolic bounds

Verified locator

DOI: 10.3390/math9030217

Source: https://arxiv.org/abs/2012.05666

Pinned version: https://arxiv.org/abs/2012.05666v1, §4.1, Conjecture 4.5. The journal version is Mathematics 9(3) (2021), 217, where the statement is Conjecture 4.

Mathematical correspondence

The defining series is

$$E_{\alpha ,m,l}(z)=\sum _{n=0}^{\infty }{c}_n{z}^n,c_n=\prod _{k=0}^{n-1}\frac{\Gamma (1+\alpha (km+l))}{\Gamma (1+\alpha (km+l+1))},c_0=1.$$

For $0<\alpha \le 1$, $m>0$, $l>m-1/\alpha$ and $x\ge 0$, the statement is

$$\frac{1}{1+\frac{\Gamma (1+\alpha (l-m))}{\Gamma (1+\alpha (l-m+1))}x}\le E_{\alpha ,m,l}(-x)\le \frac{1}{1+\frac{\Gamma (1+\alpha l)}{\Gamma (1+\alpha (1+l))}x}.$$

Defs.ksCoeff, Defs.kilbasSaigo and Defs.claim in D5/S3/Analytic/KilbasSaigo/Defs encode exactly these objects. D5/S3/Analytic/KilbasSaigo/Result.result proves the displayed bounds with the complete parameter range. The source records $l=m-1$ and $m=1$ as known cases. The boundary $l=m-1/\alpha$ belongs to the distinct Conjecture 3.7 in the pinned preprint.

The bounded literature scope and its limitations are stated in Problems/boudabsa-simon-2021-kilbas-saigo-hyperbolic-bounds.md.