While solving recursion $p_i = p_{i-B} (1-\alpha) + p_{i+1} \alpha$, related to a special gambler ruin problem, we came across the following summation. Let $0$ and $K$ be the extremes of the gambler ruin problem, that gains 1 per round with probability $\alpha$ and loses $B$ with probability $1-\alpha$. Is there a way to simplify this sum of binomial coefficients?
$$p_i = \kappa \sum\limits_{l=0}^{m} \beta^l \binom{l(B+1)-i}{l}$$ where $\beta= \alpha^{B}(1-\alpha)$ is a number between 0 and 1, and $i$ is a number between 0 and $K$. Here $\kappa$ is a constant that depends on $i$, $K$ and $B$. Finally, $m= \lfloor (i-1)/(B+1) \rfloor$.
Initial conditions: $p_0=0$ and $p_K=1$.
More precisely,
$$p_i = \frac{ \alpha^{1-i} \sum\limits_{l=0}^{\lfloor (i-1)/(B+1)\rfloor} \beta^l \binom{l(B+1)-i}{l}}{ \alpha^{1-K} \sum\limits_{l=0}^{\lfloor (K-1)/(B+1)\rfloor} \beta^l \binom{l(B+1)-i}{l}}$$
We let $\binom{n}{l}\triangleq\frac{(-1)^l (-n)_l}{l!}$ with $(n)_l$ being the rising factorial of $n$, $(n)_l=n(n+1)\cdots(n+l-1)$. NIST Handbook Mathematical Functions: what is the binomial coefficient involving negative elements? what is the terminology for the rising factorial?
If $B=1$
$$p_i = \frac{ \alpha^{1-i} \sum\limits_{l=0}^{\lfloor (i-1)/2\rfloor} \beta^l \binom{2l-i}{l}}{ \alpha^{1-K} \sum\limits_{l=0}^{\lfloor (K-1)/2\rfloor} \beta^l \binom{2l-i}{l}} = \frac{1-((1-\alpha)/\alpha)^i}{1-((1-\alpha)/\alpha)^K}$$
How to verify the latter equality? How to generalize it for the case where $B>1$?