Multiple Legendre polynomials in Diophantine approximation

We construct a class of multiple Legendre polynomials and prove that they satisfy an Ap\'ery-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree. We prove e.g. that the nonquadraticity exponent of $\log 2$ is bounded from above by $12.841618...$, thus improving upon a recent result of the author. Our construction also yields some other known results.