Generating functions for multiple zeta star values

Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 2, pp. 343-360.

Nous étudions les fonctions génératrices des valeurs des fonctions polyzêta ${\zeta }^{*}(s_1,...,s_m)$ dans le cadre général. Ces fonctions génératrices établissent un lien entre les nombres polyzêta et les sommes d’Euler multiples, ce qui nous permet d’exprimer chaque valeur polyzêta en termes de sommes d’Euler multiples alternées, et notamment réduire la longueur des blocs de deux dans les sommes résultantes.

We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value in terms of multiple alternating Euler sums, and specifically, reduce the length of blocks of twos in the resulting sums.

Classification : 11M32, 11M35, 05A15, 30B10, 30D05, 39B32
Mots clés : Multiple zeta star value, multiple zeta value, generating function, Euler sum

[1] Borwein, Jonathan M.; Bradley, David M.; Broadhurst, David J.; Lisoněk, Petr Special values of multiple polylogarithms, Trans. Am. Math. Soc., Volume 353 (2001) no. 3, pp. 907-941 | DOI | MR | Zbl

[2] Brown, Francis Mixed Tate motives over $\mathbb{Z}$, Ann. Math., Volume 175 (2012) no. 2, pp. 949-976 | DOI | MR | Zbl

[3] Flajolet, Philippe; Salvy, Bruno Euler sums and contour integral representations, Exp. Math., Volume 7 (1998) no. 1, pp. 15-35 | DOI | MR | Zbl

[4] Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana An alternative proof of a theorem of Zagier, J. Math. Anal. Appl., Volume 449 (2017) no. 1, pp. 168-175 | DOI | MR | Zbl

[5] Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana Multiple zeta star values on 3-2-1 indices (2018) (https://arxiv.org/abs/1806.10510)

[6] Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana; Tauraso, Roberto New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner’s series, Trans. Am. Math. Soc., Volume 366 (2014) no. 6, pp. 3131-3159 | MR | Zbl

[7] Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana; Zhao, Jianqiang On $q$-analogs of some families of multiple harmonic sums and multiple zeta star value identities, Commun. Number Theory Phys., Volume 10 (2016) no. 4, pp. 805-832 | DOI | MR | Zbl

[8] Hoffman, Michael E. The algebra of multiple harmonic series, J. Algebra, Volume 194 (1997) no. 2, pp. 477-495 | DOI | MR | Zbl

[9] Linebarger, Erin; Zhao, Jianqiang A family of multiple harmonic sum and multiple zeta star value identities, Mathematika, Volume 61 (2015) no. 1, pp. 63-71 | DOI | MR | Zbl

[10] Ohno, Yasuo; Zudilin, Wadim Zeta stars, Commun. Number Theory Phys., Volume 2 (2008) no. 2, pp. 325-347 | DOI | MR | Zbl

[11] Zagier, Don Evaluation of the multiple zeta values $\zeta (2,...,2,3,2,...,2)$, Ann. Math., Volume 175 (2012) no. 2, pp. 977-1000 | DOI | MR | Zbl

[12] Zhao, Jianqiang Identity families of multiple harmonic sums and multiple zeta star values, J. Math. Soc. Japan, Volume 68 (2016) no. 4, pp. 1669-1694 | DOI | MR | Zbl