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

Zudilin, Wadim

J. Comput. Appl. Math. 202, No. 2, 450-459 (2007).

Using hypergeometric series, simultaneous approximations for polylogarithms are proposed of the form \(r_n(z)=a_n \text{Li}_1(z)-b_n\) and \(\widetilde{r}_n(z)= a_n \text{Li}_2(z)-\widetilde{b}_n\) where \(a_n\) is a polynomial in \(1/z\) and \(b_n\) and \(\widetilde{b}_n\) are sums of polynomials in \(1/z\) and \(z/(z-1)\). By analytic continuation, this gives simultaneous approximations to \(\text{Li}_1(-1)\) and \(\text{Li}_2(-1)\) in which case Apéry-like recurrence relations of order 3 for \(a_n, b_n\) and \(\widetilde{b}_n\), and hence also for \(r_n\) and \(\widetilde{r}_n\) are obtained.
Two generalizations are given. The first is also including \(\widetilde{\widetilde{r}}_n(z)=a_n\text{Li}_3(z)-\widetilde{\widetilde{b}}_n\), giving approximations for \(z=1\) to \(\zeta(2)\) and \(\zeta(3)\), and as before, recurrence relations for the \(a_n\), \(\widetilde{b}_n\), \(\widetilde{\widetilde{b}}_n\), \(\widetilde{r}_n\) and \(\widetilde{\widetilde{r}}_n\). The second generalization introduces well-poised hypergeometric series, which leads for \(z=-1\) to simultaneous approximations to the numbers \(\pi^2/12\) and \(3\zeta(2)/2\).

Reviewer: Adhemar Bultheel (Leuven)

MSC:

65D20	Computation of special functions and constants, construction of tables
33C20	Generalized hypergeometric series, \({}_pF_q\)
33F10	Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
11J70	Continued fractions and generalizations
11M06	\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

polylogarithm; dilogarithm; trilogarithm; zeta function; hypergeometric series; Apéry approximation

References:

[1]	Apéry, R., Irrationalité de \(\zeta(2)\) et \(\zeta(3)\), Astérisque, 61, 11-13 (1979) · Zbl 0401.10049
[3]	Ball, K.; Rivoal, T., Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math., 146, 1, 193-207 (2001) · Zbl 1058.11051
[4]	Beukers, F., A note on the irrationality of \(\zeta(2)\) and \(\zeta(3)\), Bull. London Math. Soc., 11, 3, 268-272 (1979) · Zbl 0421.10023
[5]	Chudnovsky, G. V., Padé approximations to the generalized hypergeometric functions. I, J. Math. Pures Appl. (9), 58, 4, 445-476 (1979) · Zbl 0434.10023
[6]	Galochkin, A. I., Lower estimates of linear forms in values of certain \(G\)-functions, Mat. Zametki [Math. Notes], 18, 4, 541-552 (1975) · Zbl 0319.10039
[7]	Hata, M., On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. (9), 69, 2, 133-173 (1990) · Zbl 0712.11040
[8]	Hata, M., Rational approximations to the dilogarithm, Trans. Amer. Math. Soc., 336, 1, 363-387 (1993) · Zbl 0768.11022
[10]	Maier, W., Potenzreihen irrationalen Grenzwertes, J. Reine Angew. Math., 156, 93-148 (1927) · JFM 53.0340.02
[11]	Nesterenko, Yu. V., Integral identities and constructions of approximations to zeta-values, J. Théorie Nombres Bordeaux, 15, 2, 535-550 (2003) · Zbl 1090.11047
[12]	Nikishin, E. M., On irrationality of values of functions \(F(x, s)\), Mat. Sb. [Russian Acad. Sci. Sb. Math.], 109, 3, 410-417 (1979) · Zbl 0414.10032
[13]	Petkovšek, M.; Wilf, H. S.; Zeilberger, D., \(A = B (1996)\), A.K. Peters, Ltd.: A.K. Peters, Ltd. Wellesley, MA
[15]	Rhin, G.; Viola, C., The permutation group method for the dilogarithm, Ann. Scuola Norm. Sup. Pisa (5), 4, 389-437 (2005) · Zbl 1170.11316
[17]	Zudilin, W., A third-order Apéry-like recursion for \(\zeta(5)\), Mat. Zametki [Math. Notes], 72, 5, 733-737 (2002) · Zbl 1041.11057