Strong asymptotics of the Hermite–Padé approximants for a system of Stieltjes functions with Laguerre weight

The Hermite–Padé approximants with common denominator are considered for a pair of Stieltjes functions with weights $x^\alpha e^{-\beta_1x}$ and $x^\alpha e^{-\beta_2x}$, where $\alpha>-1$, $\beta_2>\beta_1>0$. On the basis of the method of the Riemann–Hilbert matrix problem the strong asymptotics of these approximants are found in the case $\beta_2/\beta_1<3+2\sqrt2$. The limiting distribution of the zeros of the denominators of the Hermite–Padé approximants is shown to be equal to the equilibrium measure of a certain Nikishin system.