Question

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

$\frac{\mathbf{\Gamma }\left({\displaystyle \frac{2}{3}}\right)}{\mathbf{\Gamma }\left({\displaystyle \frac{5}{3}}\right)}$

Short answer

The value of $\frac{\Gamma \left({\displaystyle \frac{2}{3}}\right)}{\Gamma \left(\frac{5}{3}\right)}$is $\frac{3}{2}$ .

Step-by-step solution

Given information

The given term to simplify is$\frac{\Gamma \left({\displaystyle \frac{2}{3}}\right)}{\Gamma \left(\frac{5}{3}\right)}$.

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.

Apply the formula

Solve the expression.

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

The value of given term is$\begin{array}{rcl}& & \frac{3}{2}\end{array}$