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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 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.

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up vote 3 down vote accepted

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.

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Have you tried to look at the recurrence relation satisfied by the product of Jacobi polynomials? (For example, maple/gfun finds product recurrences, but a list of related packages is on the site of the Seminaire Lotharingien de Combinatoire.)

In general, this will not be a single sum, but a lot of things can be done just by using the recurrence relation instead of a sum representation and there is the Petkovsek algorithm to check if a nice closed-form solution exists.

(A good reference is the book A=B by Petkovsek, Wilf and Zeilberger

(BTW I don't think that your expression for the coefficient is correct but I may have copied it incorrectly.)

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