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Sequence operator and transform

  1. Mar 4, 2008 #1
    Define a shift polynomial sequence operator as
    [tex]S=\sum_{i=0}^mc_i\mathbf E^i[/tex]

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

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

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

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

  2. jcsd
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