Q: Orthogonal relation between divided functions. Let $m$ and $n$ are two integers such that , 


*

*If $m \neq n$  then  $$\int _a^b {\dfrac{f_n(x)}{f_m(x)} 
 \ dx}=0$$

*If $m=n$  then  $$\int _a^b {\dfrac{f_n(x)}{f_n(x)} \ dx}=\int _a^b 1 \ dx=b-a$$


I am looking for general properties of such functions ( such as how to find their differential equation, generating functions, polynomials, etc. ) 
For now I have found one example which is 
$$ f_n(x)= e^{2\pi inx}\ \rm{for} \ a=0, b=1 .$$  
How can we find the general properties of such functions or other functions that satisfy the above property ?
Thank you for answers. 
EDIT: We can find a function expression via using  $f_n(x)$.
$$g(x)= \cdots +a_{-2}f_{-2}(x)+a_{-1}f_{-1}(x)+a_0f_0(x)+a_1f_1(x)+a_2f_2(x)+\cdots $$
$$ \int _a^b {\dfrac{g(x)}{f_m(x)} \ dx} =a_m (b-a)$$
or
$$g(x)= \cdots + \frac{b_{-2}}{f_{-2}(x)}+ \frac{b_{-1}}{f_{-1}(x)}+ \frac{b_{0}}{f_{0}(x)}+\frac {b_{1}}{f_{1}(x)}+\frac {b_{2}}{f_{2}(x)}+\cdots$$
$$ \int _a^b g(x)f_m(x)  dx=b_m (b-a)$$
Thus
$$\frac{a_m}{b_m}=\frac{\int _a^b {\dfrac{g(x)}{f_m(x)} \ dx} }{\int _a^b g(x)f_m(x)  dx}$$
Maybe it can be helpful further analysis to find such functions.
