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Exact values of Z_β(α) for small positive zeros of E_{α,β}(-z^α)

Determine the exact values of Z_β(α), defined for β ∈ {1, α, 2} and α ∈ (1,2] as the smallest positive real number z such that E_{α,β}(−z^α) = 0, thereby providing analytic results for the small-modulus zeros of these Mittag-Leffler functions.

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Background

To apply the separation results quantitatively, one needs analytic information about small positive zeros of E_{α,β}(−zα). The authors note that while asymptotics for large-modulus zeros are available in the literature, small-modulus results appear to be missing.

They introduce Z_β(α) as the smallest positive zero and provide numerical studies, but call for analytic determination of these values.

References

While a large amount of information is available on the zeros with large modulus of these functions (see, e.g., [15, Section 4.6]), we have unfortunately been unable to find any concrete statements about the zeros with small modulus. ... We hope that our paper will initiate further analytical investigations on the question for the exact values of Z_ (α).

— 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 (Zeros of Mittag-Leffler functions)