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

Using hypergeometric series, simultaneous approximations for polylogarithms are proposed of the form \(r_n(z)=a_n \text{Li}_1(z)-b_n\) 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 \(3\zeta(2)/2\).


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 \(L(s, \chi)\)
Full Text: DOI arXiv


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.