backshift operator in time series confusion Hi All: In time series econometrics, if one has a relationship say

$y_{t} = \frac{x_{t}}{1- \rho L} + \epsilon_{t} $ where L is the backshift operator, then the expression can be re-written as the following infinite series:

$y_{t} = \sum_{i=0}^{\infty} \rho^{i} x_{t-i} + \epsilon_{t}$.

But, what is the expression ( and can it be derived easily ) when the original expression is

$y_{t} = \frac{x_{t}}{1- \rho (L)^{k}}$ where $k is some positive integer greater than 1. Just to be clear, since this is coming out so small when I look at it, the rho is not raised to k but the backshift operator is.

Thanks for the wisdom.  Also, if it's too complicated  or too long to explain, then any reference that explains it is appreciated.
From your last comment it seems that you might have already figured out how to proceed, although I'm not following some of your reasoning. I just want to point out that there is no need to initialize $y_0$.

The absolute value of ρ is typically assumed to be lower than 1 (otherwise, the series would be "explosive"). Because of that, the terms $ρ^ny_{t-n*k}$ and $ρ^nϵ_{t-n*k}$ will be zero when taking the limit of $n-&gt;∞$ in the expansion of $y_t$.

The expression for $y_t$ is 
$$ y_t= ∑_{i=0}^{∞}  ρ^ix_{t−i*k}+ϵ_t.$$

This means that the effect of $ρL^k$ is to model $y_t$ from $x_t$, $x_{t-k}$, $x_{t-2k}$, $x_{t-3k}$, and so forth. $k=1$ is just a special case of this expression.

I feel too lazy to plug my scanner and scan my derivation right now, but I'll do so if you really want to see it.

Edited to add scan and explanation of the derivation

Here's a walk-through of the derivation. Hope my handwriting is legible enough; if not, it's the scanner's fault. 


(1) shows the equivalent way of writing $y_t$ by applying the
backshift operator $L^k$ (of course, after multiplying everything by $1-ρL^k$ ).
(2) and (3) apply the backshift operator to $y_{t-k}$ and
$y_{t-2*k}$, since each one appears in each one's preceding equation.
(4) substitutes in (1) the expressions obtained from (2) and (3).
Multiplications by ρ are developed in (5). Notice the resulting pairs of
(underlined) terms offsetting each other.
(6) is (5) after removing the underlined/offsetting terms.
The pattern is discernible from (6) and expressed in (7) for $n$.
(8) is (7) but with summation notation for the terms involving $x$.
(9) is the limit of (8) when $n$ tends to infinity. There, the
assumption of |ρ|&lt;1 is essential for convergence of the summation
(as justified in the 2nd paragraph I wrote this morning).


Since $|ρ|&lt;1$, then $|ρ^n|&lt;&lt;1$ for sufficient large $n$. Therefore, the term $ρ^ny_{t-nk}$ in (8) will tend to zero regardless of the value of $y_0$. In other words, the influence of $y_0$ on $y_t$ decreases as $t$ departs from 0.