A New Shuffle Convolution for Multiple Zeta Values
Ae Ja Yee
Department of Mathematics The Pennsylvania State University University Park PA 16802
DOI: 10.1007/s10801-005-6280-y
Abstract
Recently, interest in shuffle algebra has been renewed due to their connections with multiple zeta values. In this paper, we prove a new shuffle convolution that implies a reduction formula for the multiple zeta value z({5,1} n).
Pages: 55–69
Keywords: keywords multiple zeta values; Euler sums; shuffle algebra; multisets
Full Text: PDF
References
1. J.M. Borwein, D.M. Bradley, and D.J. Broadhurst, “Evaluations of k-fold Euler/Zagier sums: A compendium of results for arbitrary k,” Elec. J. Comb. 4(2) (1997), #R5.
2. J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lison\check ek, “Combinatorial aspects of multiple zeta values,” Elec. J. Comb. 5(1) (1998), #R38.
3. J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lison\check ek, “Special values of multiple polylogarithms,” Trans. Amer. Math. Soc. 353(3) (2000), 907-941.
4. D. Bowman and D.M. Bradley, “The algebra and combinatorics of shuffles and multiple zeta values,” J. Combin. Theory Ser. A 97(1) (2002), 43-61.
5. D. Bowman and D.M. Bradley, “Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth,” Compositio Mathematica 139(1) (2003), 85-100.
6. D. Bowman and D.M. Bradley, “Multiple polylogarithms: A brief survey,” in q-series with Applications to Combinatorics, Number Theory and Physics, B.C. Berndt and K. Ono (Eds.), Contemp Math. American Mathematical Society, Providence, RI, 2001, pp. 71-92.
7. D. Bowman, D.M. Bradley, and J.H. Ryoo, “Some multi-set inclusions associated with shuffle convolutions and multiple zeta values,” European J. Combinatorics 24 (2003), 121-127.
8. D.J. Broadhurst and D. Kreimer, “Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops,” Phys. Lett. B 393(3/4) (1997), 403-412.
9. M.E. Hoffman, “Multiple harmonic series,” Pacific J. Math. 152(2) (1993), 275-290.
10. M.E. Hoffman, “The algebra of multiple harmonic series,” J. Algebra 194 (1997), 477-495.
11. D.E. Loeb, “Sets with a negative number of elements,” Adv. Math. 91 (1992), 64-74.
12. M. Waldschmidt, “Valeurs z\hat eta multiples: Une introduction,” J. Théorie des Nombres de Bordeaux 12 (2000), 581-595.
13. M. Waldschmidt, “Introduction to polylogarithms,” in Number Theory and Discrete Mathematics, A.K. Agarwal, B.C. Berndt, C. Krattenthaler, G.L. Mullen, K. Ramachandra, and W. Waldschmidt (Eds.), Hindustan Book Agency, New Delhi, 2002, pp. 1-12.
2. J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lison\check ek, “Combinatorial aspects of multiple zeta values,” Elec. J. Comb. 5(1) (1998), #R38.
3. J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lison\check ek, “Special values of multiple polylogarithms,” Trans. Amer. Math. Soc. 353(3) (2000), 907-941.
4. D. Bowman and D.M. Bradley, “The algebra and combinatorics of shuffles and multiple zeta values,” J. Combin. Theory Ser. A 97(1) (2002), 43-61.
5. D. Bowman and D.M. Bradley, “Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth,” Compositio Mathematica 139(1) (2003), 85-100.
6. D. Bowman and D.M. Bradley, “Multiple polylogarithms: A brief survey,” in q-series with Applications to Combinatorics, Number Theory and Physics, B.C. Berndt and K. Ono (Eds.), Contemp Math. American Mathematical Society, Providence, RI, 2001, pp. 71-92.
7. D. Bowman, D.M. Bradley, and J.H. Ryoo, “Some multi-set inclusions associated with shuffle convolutions and multiple zeta values,” European J. Combinatorics 24 (2003), 121-127.
8. D.J. Broadhurst and D. Kreimer, “Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops,” Phys. Lett. B 393(3/4) (1997), 403-412.
9. M.E. Hoffman, “Multiple harmonic series,” Pacific J. Math. 152(2) (1993), 275-290.
10. M.E. Hoffman, “The algebra of multiple harmonic series,” J. Algebra 194 (1997), 477-495.
11. D.E. Loeb, “Sets with a negative number of elements,” Adv. Math. 91 (1992), 64-74.
12. M. Waldschmidt, “Valeurs z\hat eta multiples: Une introduction,” J. Théorie des Nombres de Bordeaux 12 (2000), 581-595.
13. M. Waldschmidt, “Introduction to polylogarithms,” in Number Theory and Discrete Mathematics, A.K. Agarwal, B.C. Berndt, C. Krattenthaler, G.L. Mullen, K. Ramachandra, and W. Waldschmidt (Eds.), Hindustan Book Agency, New Delhi, 2002, pp. 1-12.