Generating functions for multiple zeta star values
Khodabakhsh Hessami Pilehrood1; Tatiana Hessami Pilehrood1
1 The Fields Institute for Research in Mathematical Sciences 222 College Street Toronto, Ontario M5T 3J1, Canada
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 343-360.

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.

Nous étudions les fonctions génératrices des valeurs des fonctions polyzêta ζ * (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.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1084
Classification: 11M32, 11M35, 05A15, 30B10, 30D05, 39B32
Keywords: Multiple zeta star value, multiple zeta value, generating function, Euler sum

Khodabakhsh Hessami Pilehrood 1; Tatiana Hessami Pilehrood 1

1 The Fields Institute for Research in Mathematical Sciences 222 College Street Toronto, Ontario M5T 3J1, Canada
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{JTNB_2019__31_2_343_0,
     author = {Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood},
     title = {Generating functions for multiple zeta star values},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {343--360},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {2},
     year = {2019},
     doi = {10.5802/jtnb.1084},
     zbl = {1391.11103},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1084/}
}
TY  - JOUR
AU  - Khodabakhsh Hessami Pilehrood
AU  - Tatiana Hessami Pilehrood
TI  - Generating functions for multiple zeta star values
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
SP  - 343
EP  - 360
VL  - 31
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1084/
DO  - 10.5802/jtnb.1084
LA  - en
ID  - JTNB_2019__31_2_343_0
ER  - 
%0 Journal Article
%A Khodabakhsh Hessami Pilehrood
%A Tatiana Hessami Pilehrood
%T Generating functions for multiple zeta star values
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 343-360
%V 31
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1084/
%R 10.5802/jtnb.1084
%G en
%F JTNB_2019__31_2_343_0
Khodabakhsh Hessami Pilehrood; Tatiana Hessami Pilehrood. Generating functions for multiple zeta star values. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 343-360. doi : 10.5802/jtnb.1084. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1084/

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

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

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

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

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

[6] Khodabakhsh Hessami Pilehrood; Tatiana Hessami Pilehrood; Roberto Tauraso 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] Khodabakhsh Hessami Pilehrood; Tatiana Hessami Pilehrood; Jianqiang Zhao 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] Michael E. Hoffman The algebra of multiple harmonic series, J. Algebra, Volume 194 (1997) no. 2, pp. 477-495 | DOI | MR | Zbl

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

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

[12] Jianqiang Zhao 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

Cited by Sources: