The Tribonacci Dirichlet series (Q6167191)

Tribonacci numbers \(T_n\) are the solutions of the 3rd order recurrence \[T_{n+3}=T_{n+2}+ T_{n+1} +T_{n}, \quad n\in \mathbb{N}, T_{1}=T_2=1, T_3=2. \;\;\;\;\; (1)\] As is well-known, they are generated by \[\sum_{n=0}^{\infty}T_nz^n=\frac{z}{1-z-z^2-z^3}\] and admit the explicit expression. As an analogue of the Fibonacci zeta-function \[\sum_{n=1}^{\infty}\frac{1}{F_n^s}\] studied by many authors, the author proves analytic continuation of the Tribonacci Dirichlet series \[\sum_{n=1}^{\infty}\frac{1}{T_n^s}\] to a meromorphic function \(\varphi_3(s)\) and determines the (simple) poles and residues. Also values at negative integers are obtained.