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Positivity and asymptotics for the smallest positive zero of E_{α,1}(-z^α)

Establish the conjectured properties of the two-parameter Mittag-Leffler function E_{α,1} evaluated at −z^α for α in (1,2]: specifically, prove that E_{α,1}(−z^α) > 0 for all z in [0, 1.559]; and determine the asymptotic behavior of Z_1(α), defined as the smallest positive real number z such that E_{α,1}(−z^α) = 0, namely Z_1(α) = (c ln(α − 1) + d)(1 + o(1)) as α → 1+ with c ≈ −0.275 and d ≈ 17.54, and Z_1(α) = (π/2) + c(2 − α) + o(2 − α) as α → 2− with c ≈ −0.12.

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Background

The function E_{α,1}(−zα) features prominently in the variation-of-constants formula and in upper bounds within the separation theory. Its smallest zero determines intervals of guaranteed positivity for kernel functions.

The authors’ numerical exploration suggests uniform positivity on a fixed interval and provides asymptotic forms for Z_1(α) near α → 1+ and α → 2−. Proving these claims would give analytic guarantees for parameter choices in their main theorems.

References

More precisely, our numerical calculations give rise to the following conjecture (see also Figures 3 and 4): (a) For any α ∈ (1,2], the statement E_{α, 1}(−zα) > 0 holds for all z ∈ [0, 1.559]. (b) For α → 1+, we have Z_1(α) = (c ln(α−1) + d) (1 + o(1)) with c ≈ −0.275 and d ≈ 17.54. (c) For α → 2−, we have Z_1(α) =  + c (2 − α) + o(2 − α) with c ≈ −0.12.

— On the separation of solutions to fractional differential equations of order $α\in (1,2)$ (2401.14771 - Chaudhary et al., 26 Jan 2024) in Section 3.3 (Small positive zeros of z ↦ E_{α,1}(−z^α))