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Congruences for central binomial sums and finite polylogarithms. (English) Zbl 1300.11020

The paper, whose approach is based on the computation of the values of finite polylogarithms \(\mathcal L_d(x)=\sum^{p-1}_{k=1} \frac{x^k}{k^d}\) in terms of more accessible quantities (e.g., Fermat quotients and Bernoulli numbers), is organized as follows.
The Introduction clarifies the aim to prove congruences, modulo a power of a prime \(p\), for finite sums involving the central binomial coefficients \(\binom {2k}{k}\) (explored, e.g., by A. J. van der Poorten [Math. Intell. 1, 195–203 (1979; Zbl 0409.10028)], L. Lewin (ed.) [Structural properties of polylogarithms. Providence, RI: American Mathematical Society (AMS) (1991; Zbl 0745.33009)], J. M. Borwein and M. Chamberland [Int. J. Math. Math. Sci. 2007, Article ID 19381, 10 p. (2007; Zbl 1137.33300)]) and the power series expansions for \((\arcsin z)^m\) (investigated, e.g., by B. C. Berndt [Ramanujan’s notebooks. Part I. New York etc.: Springer-Verlag (1985; Zbl 0555.10001)], W. Chu and D. Zheng [Int. J. Number Theory 5, No. 3, 429–448 (2009; Zbl 1191.05014)], J. M. Borwein et al. [Exp. Math. 10, No. 1, 25–34 (2001; Zbl 0998.11045)]). After having recalled further feature works (e.g., by S. Mattarei [J. Algebra 294, No. 1, 1–18 (2005; Zbl 1085.17003)], H. Strade [Simple Lie algebras over fields of positive characteristic. I: Structure theory. Berlin: de Gruyter (2004; Zbl 1074.17005)]), the authors explain that, letting \(p\) be a prime and \(d = 0,1,2,3,4\), their main achievement consists of congruences for the polynomials \(p \sum^{p-1}_{k=1} \frac{t^k}{k^d \binom{2k}{k}} \pmod{p^3}\) and \(p \sum^{p-1}_{k=1} \frac{H_{k-1}(2)}{k^d \binom{2k}{k}} t^k \pmod p\), whose explicit evaluations for specific values of \(t\) depend on the finite polylogarithms \(\mathcal L_d\) which satisfy functional equations and other relations described in the next Section.
The Section 2 collects basic identities for all finite polylogarithms \(\mathcal L_d\), such as the \(distribution\) and \(inversion\) relations found by P. Elbaz-Vincent and H. Gangl [Compos. Math. 130, No. 2, 161–210, 211–214 (2002; Zbl 1062.11042)], necessary in the rest of the paper. The authors refer to a procedure for deducing some of them, available in [L. Lewin, Polylogarithms and associated functions. New York, Oxford: North Holland (1981; Zbl 0465.33001)]. A notable result from the same [S. Mattarei and R. Tauraso, J. Integer Seq. 13, No. 5, Article ID 10.5.1, 12 p. (2010; Zbl 1217.11003)] is mentioned too.
Using a congruence modulo \(p^2\) of Xia Zhou and Tianxin Cai [Proc. Am. Math. Soc. 135, No. 5, 1329–1333 (2007; Zbl 1115.11006)], the Section 3 refines the proofs of functional equations (modulo \(p\)), relating \(\mathcal L_1(x)^2\) and \(\mathcal L_1(x)^3\) to values of \(\mathcal L_2\) and \(\mathcal L_3\), supplied by A. Granville [Integers 4, Paper A22, 3 p. (2004; Zbl 1083.11005)] and by K. Dilcher and L. Skula [Integers 6, Paper A24, 12 p. (2006; Zbl 1103.11011)]. The authors guess that their proofs might resemble the omitted ones by M. Mirimanoff [J. Reine Angew. Math. 128, 45–68 (1904; JFM 35.0216.03)], who was the first to deal with finite polylogarithms in this way, and they remark the massive presence of sixth order transformation groups in the studies on Fermat’s last theorem (reported, e.g., in [P. Ribenboim, 13 lectures on Fermat’s Last Theorem. New York etc.: Springer-Verlag (1979; Zbl 0456.10006)]).
The Section 4 presents the evaluations of \(\mathcal L_d(x)\) modulo \(p\), for \(d=1,2,3\), obtained via known congruences for special values of \(\mathcal L_d(x)\) and with the identities for finite polylogarithms given in the Section 2 and others from R. Tauraso and J. Zhao [J. Comb. Number Theory 2, No. 2, 129–159 (2010; Zbl 1245.11008)] and from Z. Sun [Discrete Appl. Math. 105, No. 1–3, 193–223 (2000; Zbl 0990.11008); Discrete Math. 308, No. 1, 71–112 (2008; Zbl 1138.11006)].
The Section 5 determines polynomial identities involving Dickson polynomials, which means sums expressible in terms of finite polylogarithms. The authors employ a congruence due to Zhihong Sun [J. Number Theory 128, No. 2, 280–312 (2008; Zbl 1154.11010)] pointing also out that the trigonometric versions of three equations of theirs have been provided by R. Wituła and D. Słota [Asian-Eur. J. Math. 1, No. 3, 439–448 (2008; Zbl 1168.05005)] through different proofs.
The Section 6 is devoted to the passage from the polynomial identities of the Section 5 to polynomial congruences by using a theorem by Jianqiang Zhao [J. Number Theory 123, No. 1, 18–26 (2007; Zbl 1115.11005)] and two identities by R. Tauraso [J. Number Theory 130, No. 12, 2639–2649 (2010; Zbl 1208.11027)].
After having switched the central binomial coefficients from the denominators to the numerators, the Section 7 establishes polynomial identities and congruences for sums similar to those previously developed. Thanks to the functional equations for the finite polylogarithms defined in the Section 2, the authors shift to higher moduli the evaluations supplied by Zhiwei Sun [Sci. China, Math. 53, No. 9, 2473–2488 (2010; Zbl 1221.11054)] and by Zhiwei Sun and R. Tauraso [Adv. Appl. Math. 45, No. 1, 125–148 (2010; Zbl 1231.11021)].
The Section 8 assembles the finite polylogarithms of the Sections 2, 3 and 4, and the polynomial congruences of the Sections 5, 6 and 7, producing numerical congruences, some of which conjectured by Zhiwei Sun [“Open conjectures on congruences”, arXiv:0911.5665] or even proved still by Zhiwei Sun [Sci. China, Math. 54, No. 12, 2509–2535 (2011; Zbl 1256.11011); “A new series for \(\pi^3\) and related congruences”, arXiv:1009.5375] and by Hao Pan and Zhiwei Sun [“Proof of three conjectures on congruences”, arXiv:1010.2489]. The authors observe that a polynomial identity, here applied without statement, is quoted in Kh. Hessami Pilehrood and T. Hessami Pilehrood [Adv. Appl. Math. 49, No. 3–5, 218–238 (2012; Zbl 1293.11030)].

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11B68 Bernoulli and Euler numbers and polynomials
33B30 Higher logarithm functions
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References:

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