New functional equations of finite multiple polylogarithms (Q2179719)

For classical multiple polylogarithms, it is well known that \[ \mathrm{Li}_{\underbrace{\scriptstyle 1,\dots,1}_{n}}(t)=\frac{1}{n!}\mathrm{Li}_1(t)^n,\;\text{ for all } n\geq 1. \] In this paper, the author obtains a finite analogue of the above formula for finite multiple polylogarithms of Ono-Yamamoyo type. Denote by $\mathcal{A}_{\mathbb{Z}[t]}=(\prod_p(\mathbb{Z}/p\mathbb{Z})[t])/(\bigoplus_p(\mathbb{Z}/p\mathbb{Z})[t])$. For a positive integer $r$ and an index $\mathbf{k}=(k_1,\dots, k_r)\in (\mathbb{Z}_{\geq 1})^r$, we define a finite multiple polylogarithm of Ono-Yamamoto type by \[ f^{OY}_{\mathcal{A},\mathbf{k}}(t)=\mathop{\sum{}^{\prime}}\limits_{0<l_1,\dots,l_r<p}\frac{t^{l_1+\dots+l_r}}{l_1^{k_1}(l_1+l_2)^{k_2}\dots(l_1+\dots+l_r)^{k_r}}\operatorname{mod} p \] as an element of $\mathcal{A}_{\mathbb{Z}[t]}$, where $\sum{}^{\prime}$ denotes the sum of fractions whose denominators are prime to $p$. Define $\zeta_{\mathcal{A}}^{(i)}(\mathbf{k})$ by \[ \zeta^{(i)}_{\mathcal{A}}(\mathbf{k})=\mathop{\sum{}^{\prime}}\limits_{\substack{0<l_1,\dots,l_r<p\\ (i-1)p<l_1+\dots+l_r<ip}}\frac{1}{l_1^{k_1}(l_1+l_2)^{k_2}\dots(l_1+\dots+l_r)^{k_r}}\;\mathrm{mod}\;p. \] The main theorem of this paper is: Theorem. For a positive integer $n$, we define two elements \[ f_n(t):=\sum_{k=0}^{n-2}\left(\sum_{i=1}^{n-k-1}\zeta_{\mathcal{A}}^{(i)}(\{1\}^{n-k-2},2)t^{i \mathbf{p}} \right)f_{\mathcal{A},\{1\}^k}^{OY}(t) \] and \[ g_n(t):=\sum_{k=0}^{n-2}\left(\sum_{i=1}^{n-k-2}\zeta_{\mathcal{A}}^{(i)}(\{1\}^{n-k-2})t^{i \mathbf{p}} \right)f_{\mathcal{A},(2,\{1\}^k)}^{OY}(t) \] of $\mathcal{A}_{\mathbb{Z}[t]}$. Here, we understand that these elements are equal to $0$ if the sums are empty. Then we have \[ f_{\mathcal{A},\{1\}^n}^{OY}(t)=\frac{1}{n!}f_{\mathcal{A},1}^{OY}(t)^n+\frac{1}{n!}\sum_{k=1}^n(k-1)!(f_k(t)+g_k(t))f_{\mathcal{A},1}^{OY}(t)^{n-k}. \] By using the above theorem, the author also obtains a functional equation for $f_{\mathcal{A},{\{1\}}^n}(t)$.