# Sequence operator and transform

1. Mar 4, 2008

### bqllpd

Define a shift polynomial sequence operator as
$$S=\sum_{i=0}^mc_i\mathbf E^i$$

where $$\mathbf E$$ is the shift operator and $$c_i$$ are some constants, variables, functions, etc. When $$S$$ is applied to a sequence $$\{a_n\}$$, then $$S(a)_n=\sum_{i=0}^mc_i\mathbf E^ia_n$$.

If $$S$$ is composed with itself $$k$$ times with the sequence, then define
$$S^k(a)_n=\left[\sum_{i=0}^mc_i\mathbf E^i\right]^ka_n$$.

If the first elements from each new sequence for each $$k$$ are taken, define this as $$S^n(a)_0=b_n=T(a)_n$$. This is a problem I'm having. I know what $$T^k(a)_n=b_n$$ and $$T^{-k}(b)_n=a_n$$ are when $$m=1$$, $$c_0=\pm k$$ when $$c_1=1$$ and $$c_0=\pm 1$$ when $$c_1=-1$$.

I want to find a general formula for $$T^k(a)_n=b_n$$ and it's inverse, $$T^{-k}(b)_n=a_n$$ where $$S$$ is any shift polynomial operator. Any help would be greatly appreciated.

Thanks
bq