9 problems

Let $0<\alpha\leq\frac{1}{2}$ and define … Here $E_{\rho,\beta}$ denotes the Mittag-Leffler function with parameters $\rho$ and $\beta$. Complete-monotonicity conjecture. The funct…

Let calpha ainleft(\frac{3}{5}, 1\right). The Mittag-Leffler function $E{\alpha,1}(z)$ is defined by its standard power series, and the imaginary axis is the set…

The parameter-boundary conjecture. There exists an increasing convex function $f:[2,\infty)\to[3,\infty)$ with $f(2)=3$ such that, for every $\alpha\ge2$,

Smallest-zero conjecture for $E_{\alpha,1}$. For any $\alpha\in(1,2]$, $E_{\alpha,1}(-z^\alpha)>0$ for all $z\in[0,1.559]$. Moreover, as $\alpha\to1+$,

Smallest-zero conjecture for $E_{\alpha,2}$. For each $\alpha\in[\alpha_0,2]$, the equation

Smallest-zero conjecture for $E_{\alpha,\alpha}$. For any $\alpha\in(1,2]$, $E_{\alpha,\alpha}(-z^\alpha)>0$ for all $z\in[0,2.93785]$. Moreover, as $\alpha\to1+$,

For $\alpha,\beta,\gamma>0$, let $F_{\alpha,\beta}^{(\gamma)}$ denote the generalized Mittag-Leffler function, and let $x>0$. Complete-monotonicity criterion conjecture. For every…

Let $\alpha\in(0,1]$, $m>0$, $l>m-1/\alpha$, and $x\geq 0$. Let $E_{\alpha,m,l}$ be the Kilbas–Saigo function. The conjectured two-sided bound. One has … The bounds are natural bec…

Let $\alpha\in(0,1]$, $m>0$, and $x\geq 0$. Define the Kilbas–Saigo function $E_{\alpha,m,l}$ by its usual power series, and let $G$ denote the Barnes-type function used in the sou…