A conjectured identity for generalized Fibonacci polynomials

Question

Let $k$ be a positive integer. Consider the generalized Fibonacci polynomials $f_{n}$ which satisfy the recursion $f_{n}=xf_{n-1}+sf_{n-k}$ with initial values $f_{n}=x^n$ for $0\le{n}<k.$ It seems that for $0\le {r,t}<k$ $$\sum_{j=0}^n {\binom{j+t-r-1}{j}s^jf_{k(n-j)+r}}=x^{r-t}f_{kn+t}.$$ Is there a simple proof of this fact?

Answer 1

In the mean-time I have seen that this result follows from the formulae $$\sum_{j} {\binom{r}{j}(-s)^j f_{n+r-kj}}=x^r f_{n}$$ and $$x^r \sum_{j} {\binom{r+j-1}{j}s^j f_{kn-kj}}=f_{kn+r},$$ which can be proved by induction.