Sum similar to the binomial sum

Question

The basic binomial theorem is $$(x+y)^n=\sum_{k=0}^n \binom nk x^ky^{n-k}\ .$$ I want to evalueate the sum similar to this.. $$\sum_{k=0}^n \binom nk x^k(1-x)^{n-k}\binom nky^k(c-y)^{n-k}\ .$$ Can this expression be further simplified ? Here, $c$ is just a scalar s.t. $c\in[0,1].$

Answer 1

Using $$\sum_{k=0}^{n} {n \choose k}^2 A ^k B^{n-k} = B^{n} ~_2F_1[-n,-n;1;\frac{A}{B}]$$ So in terms og Gauss hypergeometric funcyion $~_2F_1$, you get

$$S=\sum_{k=0}^{n} {n \choose k} x^k (1-x)^{n-l} {n \choose k} y^k (c-y)^{n-k}$$ $$\implies S=\sum_{k=0}^{n} {n \choose k}^2 (x)^k [(1-x)(c-y)]^{n-k}.$$ $$\implies S=[(1-x)(c-y)]^n ~ ~_2F_1 \left[-n,-n;1;\frac{xy}{(1-x)(c-y)}\right].$$