On the first zeros of the Mittag-Leffler function 𝐸_{𝛼,𝛼}(𝑧) for 𝛼>1

Source abstract

We investigate the first zeros (roots) of the Mittag-Leffler function by constructing and analyzing the eigenvalues of the associated fractional spectral problem. By extending special cases from previous studies, we establish a series of new results, including: (1) the function E α , α ( − z ) E_{\alpha ,\alpha }(-z) has no zeros within the disk centered at the origin with radius π 2 Γ ( α − 1 ) \pi ^2\Gamma (\alpha -1) in the complex plane; (2) let λ ( α ) \lambda (\alpha ) be the first zero (i.e., the zero with the smallest modulus) of E α , α ( − z ) E_{\alpha ,\alpha }(-z) , which always exists and is real, then the ratio Γ ( α − 1 ) λ ( α ) \frac {\Gamma (\alpha -1)}{\lambda (\alpha )} is strictly decreasing over α ∈ ( 1 , ∞ ) \alpha \in (1,\infty ) as a continuous function of α \alpha . Moreover, extended results and theoretical insights are also presented.