Limit of sum of ratios of gamma function like products

Question

I'm interested in evaluating:

$$\lim_{n \rightarrow \infty} \sum_{i=1}^n \frac{1}{i^2} [ \prod_{j=i}^n \frac{kj -1}{kj} ]^2$$

for $k$ some constant. After trying a bit, I have not figured out if the expression converges or diverges. Wolfram has given the clue that without the $1 / i^2$, the sum of products diverges.

Any other insights or connections to known expressions would be appreciated.

Answer 1

Let's first consider the infinite product$\displaystyle\lim_{n\to\infty} \prod_{j=1}^{n}\frac{kj-1}{kj}$

We know that if $1+a_j=1-(1-\frac{kj-1}{kj})>0$ then $\displaystyle\prod_{j=1}^{\infty}1+a_j$ converges if and only if $\displaystyle\sum_{j=1}^{\infty}a_j$ converges.

$a_j=1-1+\frac{1}{kj} = \frac{1}{kj}$, we notice that $\displaystyle\sum_{j=1}^{\infty}a_j$ diverges and therefore the infinite product diverges.

Please note the condition of $\frac{kj-1}{kj}>0$ for this to apply must be eventually true since it approaches 1, and the beginning terms can be factored out. Also note that this is not a complete answer, since I have not addressed anything besides the product.

Answer 2

Using first Pochhammer symbols and converting to the gamma function $$ \prod_{j=1}^{n}\frac{kj-1}{kj}=\frac{\left(i-\frac{1}{k}\right)_{n-i+1}}{(i)_{n-i+1}}=\frac{\Gamma (i)\, \Gamma \left(n+1-\frac{1}{k}\right)}{\Gamma (n+1)\, \Gamma \left(i-\frac{1}{k}\right)}$$ So, you are considering $$S_{k,n}=\frac{1}{\Gamma (n+1)^2}\sum_{j=1}^{n}\Bigg(\frac{\Gamma (i)\,\, \Gamma \left(n+1-\frac{1}{k}\right)}{i\,\, \Gamma \left(i-\frac{1}{k}\right)}\Bigg)^2$$

For $k=1$ the result is $$S_{1,n}=\frac 1 n-\frac{12 \psi ^{(0)}(n+1)+6 \psi ^{(1)}(n+1)-(\pi ^2-12 \gamma) }{6 n^2}$$ Expanded as series for large values of $n$ $$S_{1,n}=\frac 1 n-\frac{12 \log (n)-(\pi ^2-12 \gamma) }{6 n^2}-\frac{2}{n^3}+\frac{2}{3 n^4}+O\left(\frac{1}{n^5}\right)$$ which is very accurate as soon as $n>5$ (the relative error is smaller than $0.01$% for $n>7$).

When $k$ is a positive integer, the result is given in terms of hypergeometric functions. For example

$$S_{3,n}=\frac{\Gamma \left(n+\frac{2}{3}\right)^2}{\Gamma \left(\frac{2}{3}\right)^2 \Gamma (n+1)^2} \, _5F_4\left(1,1,1,1,1;\frac{2}{3},\frac{2}{3},2,2;1\right)-$$ $$\frac{\, _5F_4\left(1,n+1,n+1,n+1,n+1;n+\frac{2}{3},n+\frac{2}{3},n+2,n+2 ;1\right)}{(n+1)^2}$$

I did not obtain the asymptotics but it seems that they all converge.

What is interesting is the plot of $n S_{k,n}$ as a function of $n$.