Are there any theorems related to the product of Jacobi/Legendre Polynomials and/or Hypergeometric functions? Specifically, I'm interested in the product of ${}\_{2}F\_{1}[-n,-n+1;2;x]$ and ${}\_{2}F\_{1}[-n-1,-n+3;2;x]$ hoping to obtain it in some form ${}\_{p}F\_{q}$.


I've found some stuff in Bailey (1928,1935), but it has solutions only for some special cases. I've also obtained the coefficient of the $k$th term $\frac{x^k}{k!}$. I get (in case I didn't make any mistakes)


$\sum\_{m=0}^{k} \binom{k}{m} \binom{n}{m}\binom{n-1}{m}\binom{n+1}{k-m}\binom{n-3}{k-m} \frac{m!\,(k-m)!}{(m+1)(k-m+1)}$, but I don't quite see what to do next. 



Product of hypergeometric functions/Jacobi Polynomials
Of course, there is no general formula of the type you wanted
but a whole bunch of the formulae expressing the product of
two $\_2F\_1$ by hypergeometric (or nearly hypergeometric) means.
They are known as Orr-type theorems and can be found in
Slater's book "Generalized hypergeometric functions", Section 2.5
(there are some instances in Bailey's "Generalized hypergeometric series"
as well). A famous example of this type is Clausen's identity
$$
{}\_2F\_1(a,b;a+b+\tfrac12\mid z)^2
={}\_3F\_2(2a,2b,a+b;2a+2b,a+b+\tfrac12\mid z).
$$
In addition, you can use the contiguous relations
[Slater, Section 1.4] which allow one to produce linear 
relations between any three functions of the form
${}\_2F\_1(a+n,b+m;c+k\mid z)$ where $n,m,k\in\mathbb Z$,
as well as the transformation [Slater, Section 1.7.1]
$$
{}\_2F\_1(a,b;c\mid z)
=(1-z)^{-a}{}\_2F\_1\biggl(a,c-b;c\Bigm|\frac{z}{z-1}\biggr).
$$


I do not see however that something spectacular happens for
your particular product. In fact, the algorithm described
in the (already mentioned) book "$A=B$" decides whether the
expression $a\_k$ given by
$$
\sum\_{k=0}^\infty a\_kz^k
:={}\_2F\_1(-n,-n+1;2\mid z){}\_2F\_1(-n-1,-n+3;2\mid z)
$$
(so that each $a\_k$ is a single hypergeometric sum) can be
represented as a sum of *finite* rational terms. If this is
the case (which I really doubt), then you will have your wanted
product as a finite sum of hypergeometric functions; if not,
then this is the *proof* that you have no expression of this type.


