Nonasymptotic properties of roots of a Mittag-Leffler type functionстатья
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Аннотация:We completely solve the problem of finding the number of positive and nonnegative roots of the Mittag-Leffler type function Erho(z; mu) = Sigma(n=0)(infinity)z(n)/Gamma(mu + n/rho), rho > 0, mu is an element of C, for rho > 1 and mu is an element of R. We prove that there are no roots in the left angular sector pi/rho less than or equal to \arg z\ less than or equal to pi for rho > 1 and 1 less than or equal to mu < 1 + 1/rho. We consider the problem of multiple roots; in particular, we show that the classical Mittag-Leffler function E-n(z; 1) of integer order does not have multiple roots.