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help mo on a recursive function

I'm interested on the behaviour on a H-length circular queue, empty before i=0, equal to P(0)=a and P(i)= b Sum(j=i-1, i-H [P(j)]), that is:

P(i) = a(1+b)^i     i=0, ..., H-1

P(i) = (1+b)P(i-1) - b P(i-H)    i>= H

I got just the following results:

P(i) = a (b(i+1) - 1)/(b - 1)  *if H=1*


lim(i->Inf [ P(i) ]) = 1 / (1-b) for each H, if | b | <1

my question is: is it possible to express this function in closed form for each H?

thank you in advance,

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