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Equality of asymptotic coefficients in Z_{α}(α) for α → 1+ and α → 2−

Determine whether the constant c appearing in the asymptotic expansion Z_{α}(α) = (c ln(α − 1) + d)(1 + o(1)) as α → 1+ coincides exactly with the constant c appearing in the asymptotic expansion Z_{α}(α) = π + c(2 − α) + o(2 − α) as α → 2− for the smallest positive zero Z_{α}(α) of E_{α,α}(−z^α).

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Background

In the conjectured asymptotic description of the smallest positive zero Z_{α}(α) of E_{α,α}(−zα), numerics suggest that both the α → 1+ and α → 2− regimes involve constants close to −0.81. Whether these constants are identical is not known.

Resolving this would refine the asymptotic theory and potentially reveal deeper structural relations between the two limiting regimes for orders approaching 1 and 2.

References

Note that coefficients with values approximately equal to −0.81 occur both in the asymptotic expansion for α → 1+ and in the expansion for α → 2−. We do not know whether or not these two coefficients are exactly identical.

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