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Approximations to di- and tri-logarithms. (English) Zbl 1220.65028

Using hypergeometric series, simultaneous approximations for polylogarithms are proposed of the form r n (z)=a n Li 1 (z)-b n and r ˜ n (z)=a n Li 2 (z)-b ˜ n where a n is a polynomial in 1/z and b n and b ˜ n are sums of polynomials in 1/z and z/(z-1). By analytic continuation, this gives simultaneous approximations to Li 1 (-1) and Li 2 (-1) in which case Apéry-like recurrence relations of order 3 for a n ,b n and b ˜ n , and hence also for r n and r ˜ n are obtained.

Two generalizations are given. The first is also including r ˜ ˜ n (z)=a n Li 3 (z)-b ˜ ˜ n , giving approximations for z=1 to ζ(2) and ζ(3), and as before, recurrence relations for the a n , b ˜ n , b ˜ ˜ n , r ˜ n and r ˜ ˜ n . The second generalization introduces well-poised hypergeometric series, which leads for z=-1 to simultaneous approximations to the numbers π 2 /12 and 3ζ(2)/2.

65D20Computation of special functions, construction of tables
33C20Generalized hypergeometric series, p F q
33F10Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
11J70Continued fractions and generalizations
11M06ζ(s) and L(s,χ)