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Let $m$ and $n$ are two integers such that ,

  1. If $m \neq n$ then $\int _a^b {\dfrac{f_n(x)}{f_m(x)} \ dx}=0$
  2. 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.

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    @Mathlover : Your edit looks good !!, may be it can lead to some seminal result.2012-07-09

0 Answers 0