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Threshold and asymptotics for the smallest positive zero of E_{α,2}(-z^α)

Determine the existence and precise value of a threshold α0 ∈ (1.59911, 1.59912) such that for α ≥ α0 the equation E_{α,2}(−z^α) = 0 admits at least one positive solution; establish the asymptotic behavior of the smallest positive zero Z_2(α) defined as the least z > 0 with E_{α,2}(−z^α) = 0, including Z_2(α) = π + (2 − α) + o(2 − α) as α → 2−, Z_2(α) = Z_2(α0) + c(α − α0)^b(1 + o(1)) as α → α0+ with Z_2(α0) ≈ 5.21066, b ≈ 1/2, c ≈ −4.8, and monotonic decrease of Z_2(α) in α; and prove that for α ∈ (1, α0), E_{α,2}(−z^α) > 0 for all z ≥ 0.

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Background

For β = 2, prior literature indicates real zeros exist only above a threshold order, but the exact value is unknown. The authors’ computations suggest α0 ≈ 1.599115206 and provide evidence for asymptotic forms and monotonicity of the smallest zero Z_2(α).

A rigorous characterization of α0 and the stated asymptotics would solidify the positivity conditions required earlier for separation bounds and clarify the transition in the zero structure of E_{α,2}(−zα).

References

Conjecture. There exists some α0 ∈ (1.59911, 1.59912) such that the following statements hold: (a) For each α ∈ [α0, 2], the equation E_{α, 2}(−zα) = 0 possesses at least one solution in (0, ∞). Denoting (as in eq. (3.1) above) the smallest of these solutions by Z_2(α), we have: (i) Z_2(α) = π + (2−α) + o(2 − α) for α → 2−, (ii) Z_2(α) = Z_2(α0) + c (α − α0)b (1 + o(1)) for α → α0+ where Z_2(α0) ≈ 5.21066, b ≈ 1/2 and c ≈ −4.8, (iii) Z_2(α) is a strictly decreasing function of α. (b) For each α ∈ (1, α0), we have E_{α, 2}(−zα) > 0 for all z ≥ 0.

— 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 3.2 (Small positive zeros of z ↦ E_{α,2}(−z^α))