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I have come up with this recurrence relation in solving a problem : $$ \begin{cases} y_i = \lambda\{\sum_{m=1}^M (y_{i-1}^m)P_m + y_{i-1}P_0\} & \quad i\ge1 \\ y_0 = 1\\ \end{cases} \ $$ which $P_m$ is described as $P_m = \binom{M}{m}p^m(1-p)^{M-m}$ (Binomial Distribution); $M$ is an integer $(M\ge1)$ and p and $\lambda$ are both real numbers between $0$ and $1$: $0\le p\le1$ , $0\le \lambda\le1$.

I want to know if it can have a closed form solution or if not, find some upper bounds for $\sum_{i=0}^\infty y_i$. Does anybody know about this recurrence relation or a similar one which can possibly help me in this problem?

Mahsa Khoshnama
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  • For every $M$, $y_i=\Theta(\mu_M^i)$ where $\mu_M=\lambda(1+(M-1)p)(1-p)^{M-1}<1$. Is this the kind of estimate you are after? – Did Jul 21 '17 at 12:40

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