RATIONAL APPROXIMATIONS FOR VALUES OF DERIVATIVES OF THE GAMMA FUNCTION

RATIONAL APPROXIMATIONS FOR VALUES OF DERIVATIVES OF THE GAMMA FUNCTION TANGUY RIVOAL Abstract. The arithmetical nature of Euler's constant ? is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions an and bn such that an/bn converges sub-exponentially to ?, viewed as ???(1), where ? is the usual Gamma function. Although this is not yet enough to prove that ? 6? Q, it is major step in this direction. In this paper, we present a different, but related, approach based on simultaneous Pade approximants to Euler's functions, from which we constuct and study a new third order recurrence that produces a sequence in Q(z) whose height grows like the factorial and that converges sub-exponentially to log(z) + ? for any complex number z ? C \ (?∞, 0], where log is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to ?(s)(1) for any integer s ≥ 1. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers ? and ???(1)? 2??(1)2 = ?(2)? ?2.

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