What is the formula (non-iterative form) for the product $\prod_{i=1}^{n}((2i-1)k+1)$

Question

I am searching for a compact (non-iterative) expression/formula for the product: $\prod_{i=1}^{n}((2i-1)k+1)$. The variable k needs to be kept.

https://www.wolframalpha.com/input/?i=product+%28%282i-1%29k%2B1%29%2C+i%3D1..infinity, http://mathworld.wolfram.com/PochhammerSymbol.html — Alessio K, Dec 23 '19 at 11:25

score 0 · Accepted Answer · answered Dec 23 '19 at 11:33

0

Also, we could use the Gamma function for this. $$ \prod _{i=1}^{n} \big(\left( 2\,i-1 \right) k+1\big)={{2}^{n}{k}^{n}\frac{\Gamma \left( {\frac {2\,nk+k+1}{2k}} \right)}{ \Gamma \left({\frac {k+1}{2k}} \right) }} $$

answered Dec 23 '19 at 11:33

GEdgar

111,679

score 0 · Answer 2 · edited Dec 23 '19 at 15:05

0

$$\prod_{i=1}^n((2i-1)k+1)=\sum_{d=0}^n\sum_{S\subseteq\{1,\ldots, n\}\\|S|=d}\prod_{i\in S}(2i-1)k^d$$

edited Dec 23 '19 at 15:05

RobPratt

45,619

answered Dec 23 '19 at 14:38

Fabio Lucchini

16,054

What is the formula (non-iterative form) for the product $\prod_{i=1}^{n}((2i-1)k+1)$

2 Answers2