We show that solution to the Hermite–Padé type I approximation problem leads in a natural way subclass of solutions Hirota (discrete Kadomtsev–Petviashvili) system and its adjoint linear problem. Our result explains appearance various ingredients integrable systems theory application multiple orthogonal polynomials, numerical algorithms, random matrices, other branches mathematical physics appl...

Padé-type approximants are rational functions that approximate a given formal power series. Boutry [Numer. Algorithms, 33 (2003), pp 113–122] constructed correspond to the averaged Gauss quadrature rules introduced by Laurie [Math. Comp., 65 (1996), pp. 739–747]. More recently, Spalević 76 (2007), 1483–1492] proposed optimal rules, have higher degree of precision than corresponding with...

Linear forms in logarithms have an important role the theory of Diophantine equations. In this paper, we prove explicit [Formula: see text]-adic lower bounds for linear rational numbers using Padé approximations second kind.

1. E. P. Merkes & W. T. Scott, "Continued fraction solutions of the Riccati equation," J. Math. Anal. Appl., v. 4, 1962, pp. 309-327. MR 25 #4167. 2. W. Fair, "Padé Approximations to the Solution of the Riccati Equation," Math. Comp., v. 18, 1964, pp. 627-634. MR 29 #6630. 3. Y. L. Luke, "The Padé table and the r-method," J. Math, and Phys., v. 37, 1958, pp. 110-127. MR 20 #5558. 4. H. T. Davis...