论文标题
在与Lucas序列相关的Hurwitz型Zeta功能上
On the Hurwitz-type zeta function associated to the Lucas sequence
论文作者
论文摘要
我们研究与Lucas序列$ u = \ = \ {u_n（p，q）_ {n \ geq 0} $相关的theta函数和hurwitz-type zeta函数，由实际数字$ p，q $在$ p $和$ q $上的某些自然假设下确定的第一种类型的第一种类型的_ {n \ geq 0} $。我们将theta函数$θ_U（t）$的渐近扩展为$ t \ downarrow 0 $，并用它来获得hurwitz-type zeta zeta zeta函数$ζ_{u} {u} \ left（s，z \ right） } \ left（z+u_ {n} \ right） ^{ - s} $ to整个复杂$ s- $ plane。此外，我们在半平面$ \ re（s）\ leq 0 $中的所有极点确定$ζ_{u} \ left（s，z \ right）$的残基。
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $θ_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $ζ_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $ζ_{U}\left( s,z\right)$ at all poles in the half-plane $\Re(s)\leq 0$.
