Question

Use the recursion relation (3.4), and if needed, equation (3.2) to simplify:

$\mathit{\Gamma }\left(\frac{2}{3}\right)\mathbf{/}\mathit{\Gamma }\left(\frac{8}{3}\right)$

Short answer

The value of $\Gamma \left(\frac{2}{3}\right)/\Gamma \left(\frac{8}{3}\right)$is $\frac{9}{10}$

Step-by-step solution

Given information

The given term to simplify is

Concept and formula

A recurrence relation in mathematics is an equation that represents the nth term of a series as a function of the k preceding terms, for some fixed k known as the order of the relation.

The formula to be used is given below.

$\mathit{\Gamma }\left(p+1\right)\mathbf{=}\mathit{p}\mathit{\Gamma }\left(p\right)$

Apply the formula

Solve the expression

$\begin{array}{rcl}\Gamma \left(\frac{8}{3}\right)& =& \Gamma \left(\frac{5}{3}+1\right)\\ \Gamma \left(\frac{8}{3}\right)& =& \frac{5}{3}\Gamma \left(\frac{5}{3}\right)\\ \Gamma \left(\frac{8}{3}\right)& =& \frac{5}{3}\left(\frac{2}{3}\right)\Gamma \left(\frac{2}{3}\right)\\ \Gamma \left(\frac{2}{3}\right)/\Gamma \left(\frac{8}{3}\right)& =& \frac{9}{10}\end{array}$

The value of given term is$\begin{array}{rcl}& & \frac{9}{10}\end{array}$