Analytic properties of generalized Mordell-Tornheim multiple Lucas L-functions

Abstract

In this article, we introduce the generalized Mordell-Tornheim multiple shifted Lucas zeta functions and the generalized Mordell-Tornheim multiple Lucas L-functions associated to Dirichlet characters and study their analytic continuations. Let $\chi_1, \dots, \chi_{k}$ be the Dirichlet characters of same modulus $t \in\mathbb{N}_{\geq 2}$ and $\chi_0$ be the principal character. The generalized Mordel-Tornheim multiple shifted Lucas zeta functions and generalized Mordel-Tornheim multiple Lucas $L$- functions are respectively defined as: \begin{eqnarray*} &&\zeta_{MTU, j, k}(s_1, \dots ,s_j; s_{j+1}, \dots ,s_{k+1} \mid h_1, \dots, h_{k})\nonumber\\ &=&\sum_{m_1=0}^{\infty} \cdots \sum_{m_k=0}^{\infty}\frac{1}{U_{tm_1+h_1}^{s_1} \cdots U_{tm_j+h_j}^{s_j}U_{t(m_1+ \dots +m_j)+{h_1+ \dots +h_j}}^{s_{j+1}} \cdots U_{t(m_1+ \dots +m_k)+{h_1+\dots+ h_k}}^{s_{k+1}}}, \end{eqnarray*} and  \begin{eqnarray*} &&\mathcal{L}_{MTU, j, k}(s_1, \dots ,s_j; s_{j+1}, \dots ,s_{k+1} \mid \chi_1, \dots, \chi_{k})\nonumber\\ &=&\sum_{m_1=1}^{\infty} \cdots \sum_{m_k=1}^{\infty}\frac{\chi_1(m_1)\dots \chi_k(m_k)}{U_{m_1}^{s_1} \cdots U_{m_j}^{s_j}U_{m_1+ \dots +m_j}^{s_{j+1}} \cdots U_{m_1+ \dots +m_k}^{s_{k+1}}}, \end{eqnarray*} where $m_i, h_i \in \mathbb{N}$ for $1\leq i \leq k.$  We also compute a complete list of exact singularities and residues of these functions at poles.