Using the combinatorial descriptions of stuffle product, we obtain recursive formulas for the stuffle product of multiple zeta values and of multiple zeta-star values. Then we apply the formulas to prove several stuffle product formulas with one or two strings of $z_p$'s. We also describe how to use our formulas in general cases.
References
K.-W. Chen, Applications of stuffle product of multiple zeta values, J. Number Theory 153 (2015), 107–116. 1365.11102 10.1016/j.jnt.2015.01.003 MR3327568 K.-W. Chen, Applications of stuffle product of multiple zeta values, J. Number Theory 153 (2015), 107–116. 1365.11102 10.1016/j.jnt.2015.01.003 MR3327568
M. Eie and C.-S. Wei, Generalizations of Euler decomposition and their applications, J. Number Theory 133 (2013), no. 8, 2475–2495. 1286.11134 10.1016/j.jnt.2013.01.010 MR3045200 M. Eie and C.-S. Wei, Generalizations of Euler decomposition and their applications, J. Number Theory 133 (2013), no. 8, 2475–2495. 1286.11134 10.1016/j.jnt.2013.01.010 MR3045200
M. E. Hoffman, The algebra of multiple harmonic series, J. Algebra 194 (1997), no. 2, 477–495. 0881.11067 10.1006/jabr.1997.7127 M. E. Hoffman, The algebra of multiple harmonic series, J. Algebra 194 (1997), no. 2, 477–495. 0881.11067 10.1006/jabr.1997.7127
P. Lei, L. Guo and B. Ma, Applications of shuffle product to restricted decomposition formulas for multiple zeta values, J. Number Theory 144 (2014), 219–233. 1351.11055 10.1016/j.jnt.2014.03.020 P. Lei, L. Guo and B. Ma, Applications of shuffle product to restricted decomposition formulas for multiple zeta values, J. Number Theory 144 (2014), 219–233. 1351.11055 10.1016/j.jnt.2014.03.020
S. Muneta, Algebraic setup of non-strict multiple zeta values, Acta Arith. 136 (2009), no. 1, 7–18. 1242.11063 10.4064/aa136-1-2 MR2469941 S. Muneta, Algebraic setup of non-strict multiple zeta values, Acta Arith. 136 (2009), no. 1, 7–18. 1242.11063 10.4064/aa136-1-2 MR2469941
D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics, Vol. II (Paris, 1992), 497–512, Progr. Math. 120, Birkhäuser, Basel, 1994. 0822.11001 D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics, Vol. II (Paris, 1992), 497–512, Progr. Math. 120, Birkhäuser, Basel, 1994. 0822.11001