Closed form expression for products How can I find a closed form expression for products of the following form $$\prod_{k=1}^n (ak^2+bk+c)\space \text{?}$$
 A: As commented by r9m, the key idea is to factorize the polynomial. When this is done, you can either use  Pochhammer functions which would give $$\prod_{k=1}^n (ak^2+bk+c)=a^n \left(\frac{2 a+b-\sqrt{b^2-4 a c}}{2 a}\right){}_n \left(\frac{2 a+b+\sqrt{b^2-4
   a c}}{2 a}\right){}_n$$ or transform to Gamma functions $$\prod_{k=1}^n (ak^2+bk+c)=a^n\frac{ \Gamma \left(n+\frac{b}{2 a}-\frac{\sqrt{b^2-4 a c}}{2 a}+1\right) \Gamma
   \left(n+\frac{b}{2 a}+\frac{\sqrt{b^2-4 a c}}{2 a}+1\right)}{\Gamma
   \left(\frac{b}{2 a}-\frac{\sqrt{b^2-4 a c}}{2 a}+1\right) \Gamma \left(\frac{b}{2
   a}+\frac{\sqrt{b^2-4 a c}}{2 a}+1\right)}$$
A: The Maple code
product(a*k^2+b*k+c, k = 1 .. n);

produces
$${\frac {{a}^{n+1}}{a}\Gamma  \left( n+1-1/2\,{\frac {-b+\sqrt {-4\,ca+
{b}^{2}}}{a}} \right) \Gamma  \left( n+1-1/2\,{\frac {-b-\sqrt {-4\,ca
+{b}^{2}}}{a}} \right) \times$$ 
$$ \left( \Gamma  \left( 1-1/2\,{\frac {-b+\sqrt 
{-4\,ca+{b}^{2}}}{a}} \right)  \right) ^{-1} \left( \Gamma  \left( 1-1
/2\,{\frac {-b-\sqrt {-4\,ca+{b}^{2}}}{a}} \right)  \right) ^{-1}}
  $$
