A product of things 2

Calculus Level pending

$\prod _{n=1}^{\infty } \left(\frac{1}{n^7}+\frac{1}{n^5}+\frac{1}{n^4}+\frac{1}{n^3}+\frac{1}{n ^2}+1\right)=\frac{\cosh ^3\left(\frac{\pi \sqrt{b}}{2}\right)}{\pi ^a}$

where $a,b$ are positive integers. Submit $a+b$ .

Bonus: Let $p(x)$ be a polynomial. Find a formula for

$\prod _{n=1}^{\infty } \left(\frac{p\left(\frac{1}{n}\right)}{n^2}+1\right)$

Bonus 2: Let $p(x)$ and $q(x)$ be polynomials with $deg(p) \leq 2 + deg(q)$ . Find a formula for

$\prod _{n=1}^{\infty } \left(\frac{p(x)}{q(x)}+1\right)$

The answer is 6.

1 solution

Christopher Criscitiello
Dec 23, 2017

Hint for Bonus: Let $d$ be the degree of $p(x)$ . The formula involves the product of $d$ Gamma functions, whose arguments are the roots of a certain polynomial of degree $d$ .

For Bonus 2, write $1+\frac{p(x)}{q(x)} = \frac{P(x)}{Q(x)}$ . Then the product equals

$\frac{\prod_{\text{roots w of Q}} \Gamma(1-w)}{\prod_{\text{roots z of P}} \Gamma(1-z)}$

(prove this)

A product of things 2

The answer is 6.

1 solution

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