In this paper, by using generating functions for the Chebyshev polynomials, we have obtained the convolution formulas involving the bi-periodic Fibonacci and Lucas polynomials.

In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by $L_n\left(x\right)=bp\left(x\right)L_{n-1}\left(x\right)+q\left(x\right)L_{n-2}\left(x\right)$ (if $n$ is even) or $L_n\left(x\right)=ap\left(x\right)L_{n-1}\left(x\right)+q\left(x\right)L_{n-2}\left(x\right)$ (if $n$ is odd), with initial conditions $L_0\left(x\right)=2$, $L_1\left(x\right)=ap\left(x\right)$, where $p\left(x\right)$ and $q\left(x\right)$ were nonzero polynomials in $Q\left[x\right]$. We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials.

In this paper, we derive the asymptotic formulas $B_{r,s,t}^{\ast }(n)$ such that

$$\underset{n\to \mathrm{\infty }}{lim}\left\{{\left(\sum _{k=n}^{\mathrm{\infty }}\frac{1}{k^r(k+t{)}^s}\right)}^{-1}-B_{r,s,t}^{\ast }(n)\right\}=0,$$

where $Re(r+s)>1$ and $t\in \mathbb{C}$. It is evident that the asymptotic formulas for the inverses of the tails of both the Riemann zeta function and the Hurwitz zeta function on the half-plane $Re(s)>1$ are its corollaries. Subsequently we provide the asymptotic formulas for the Riemann zeta function and the Hurwitz zeta function on the half-plane $Re(s)<0$. Finally, we study the asymptotic formulas of the inverse of the tails of the Dirichlet L-function for $Re(s)>1$ and $Re(s)<0$.