Closed form expression for products

Question

How can I find a closed form expression for products of the following form $$\prod_{k=1}^n (ak^2+bk+c)\space \text{?}$$

Answer 1

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 $$\prod_{k=1}^n (ak^2+bk+c)=a^n\frac{ \Gamma \left(n+\frac{b}{2 a}-\frac{\sqrt{b^2-4 a c}}{2 a}+1\right) \Gamma \left(n+\frac{b}{2 a}+\frac{\sqrt{b^2-4 a c}}{2 a}+1\right)}{\Gamma \left(\frac{b}{2 a}-\frac{\sqrt{b^2-4 a c}}{2 a}+1\right) \Gamma \left(\frac{b}{2 a}+\frac{\sqrt{b^2-4 a c}}{2 a}+1\right)}$$

Answer 2

The Maple code

product(a*k^2+b*k+c, k = 1 .. n);

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