Geodesic
Strong asymptotics of the Hermite–Padé approximants for a system of Stieltjes functions with Laguerre weight
Sbornik. Mathematics, Tome 196 (2005) no. 12, pp. 1815-1840 
V. G. Lysov. Strong asymptotics of the Hermite–Padé approximants for a system of Stieltjes functions with Laguerre weight. Sbornik. Mathematics, Tome 196 (2005) no. 12, pp. 1815-1840. http://geodesic.mathdoc.fr/item/SM_2005_196_12_a3/
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     title = {Strong asymptotics of the {Hermite{\textendash}Pad\'e} approximants for a system of {Stieltjes} functions with {Laguerre} weight},
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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.

[1] Gonchar A. A., Rakhmanov E. A., “O skhodimosti sovmestnykh approksimatsii Pade dlya sistemy funktsii markovskogo tipa”, Trudy MIAN, 157, 1981, 31–48 | MR | Zbl

[2] Nikishin E. M., Sorokin V. N., Ratsionalnye approksimatsii i ortogonalnost, Nauka, M., 1988 | MR | Zbl

[3] Aptekarev A. I., Branquinho A., Van Assche W., “Multiple orthogonal polynomials for classical weights”, Trans. Amer. Math. Soc., 355 (2003), 3887–3914 | DOI | MR | Zbl

[4] Sorokin V. N., “Skhodimost sovmestnykh approksimatsii Pade k funktsiyam stiltesovskogo tipa”, Izv. vuzov. Ser. matem., 1987, no. 7, 48–56 | MR | Zbl

[5] Nuttall J., “Asymptotics of diagonal Hermite–Padé polynomials”, J. Approx. Theory, 42:4 (1984), 299–386 | DOI | MR | Zbl

[6] Deift P., Zhou X., “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for MKdV equation”, Ann. of Math. (2), 137:2 (1993), 295–368 | DOI | MR | Zbl

[7] Van Assche W., Geronimo J. S., Kuijlaars A. B. J., “Riemann–Hilbert problems for multiple orthogonal polynomials”, Special functions 2000: Current perspective and future directions (Tempe, AZ), eds. J. Bustoz et al., Kluwer Acad. Publ., Dordrecht, 2001, 23–59 | MR | Zbl

[8] Gonchar A. A., Rakhmanov E. A., Sorokin V. N., “Ob approksimatsiyakh Ermita–Pade dlya sistem funktsii markovskogo tipa”, Matem. sb., 188:5 (1997), 33–58 | MR | Zbl

[9] Kuijlaars A. B. J., McLaughlin K. T.-R., Van Assche W., Vanlessen M., “The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$”, Adv. Math., 188 (2004), 337–398 | DOI | MR | Zbl

[10] Aptekarev A. I., Van Assche W., “Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight”, J. Approx. Theory, 129 (2004), 129–166 | DOI | MR | Zbl

[11] Kuijlaars A. B. J., Van Assche W., Wielonsky F., “Quadratic Hermite–Padé approximation to the exponential function: a Riemann–Hilbert approach”, Constr. Approx., 21:3 (2005), 351–412 | DOI | MR | Zbl