By using the method of iterated integral representations of series, weestablish some explicit relationships between multiple zeta values andIntegrals of logarithmic functions. As applications of these relations, we showthat multiple zeta values of the form [zeta ( {ar 1,{{left{ 1ight}}_{m - 1}},ar 1,{{left{ 1 ight}}_{k - 1}}} ), (k,minmathbb{N})] for $m=1$ or $k=1$, and [zeta ( {ar 1,{{left{ 1ight}}_{m - 1}},p,{{left{ 1 ight}}_{k - 1}}}), (k,minmathbb{N})]for $p=1$ and $2$, satisfy certain recurrence relations which allow us to writethem in terms of zeta values, polylogarithms and $ln 2$. Moreover, we alsoprove that the multiple zeta values $zeta ( {ar 1,{{left{ 1 ight}}_{m -1}},3,{{left{ 1 ight}}_{k - 1}}} )$ can be expressed as a rational linearcombination of products of zeta values, multiple polylogarithms and $ln 2$when $m=kin mathbb{N}$. Furthermore, we also obtain reductions for certainmultiple polylogarithmic values at $rac {1}{2}$.
展开▼
