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May 24th, 2019, 08:53 AM   #1
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Functions in function - synthesis of a general form

Hi All,

I've one problem related to defining general form of fuction f(x) which is constructed by using functions inside of it. Let we say, that f(x) is defined as follows:
$\displaystyle f(x)= \frac{c \cdot a_{0}+ f_{1}(x) }{ a_{0}+ c \cdot f_{1}(x)} $
and the two next functions are:
$\displaystyle f_{1}(x)= \frac{c \cdot a_{1}+ f_{2}(x) }{ a_{1}+ c \cdot f_{2}(x)} $, $\displaystyle f_{2}(x)= \frac{c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} $ ..., $\displaystyle f_{n}(x)= \frac{c \cdot a_{n}+ f_{n+1}(x) }{ a_{n}+ c \cdot f_{n+1}(x)} $

Firstly, I thought that this can be made by using recursive theory, however this function is not "calling itself". Once again - to be clear - I want to find general form of function f(x) which will be related to the number of functions constructing it, and furthermore - it will have rational function character (so it will have polynomials in a numerator, and in a denominator).

I believe, that there are some methods to solve this problem but I cannot find them. Could you help me, and give some suggestions how can I figure it out ?

Best Regards,
E.
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May 24th, 2019, 01:04 PM   #2
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Your construction looks very difficult in that there is no starting point.
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May 24th, 2019, 02:40 PM   #3
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Ok, so I'll try to start from the beginning.

In last time, I was thinking about what will happen If I take simple function, and replace some parameter in it by using some other or the same function. So in the first step, I defined function f(x):
$\displaystyle f(x)= \frac{c \cdot a_{0}+ b }{ a_{0}+ c \cdot b} $
In this formula I've used parameter b, and replaced it by using function f1(x):
$\displaystyle f_{1}(x)= \frac{c \cdot a_{1}+ f_{2}(x) }{ a_{1}+ c \cdot f_{2}(x)} $

after that, my idea was to find a new, general form of function f(x), by substituting f1(x) to the previous f(x), i.e.:
$\displaystyle f(x)= \frac{ c \cdot a_{0} + ( \frac{ c \cdot a_{1}+ f_{2}(x) }{ a_{1}+ c \cdot f_{2}(x)} ) }{ a_{0} + c \cdot ( \frac{ c \cdot a_{1}+ f_{2}(x) }{ a_{1}+ c \cdot f_{2}(x)} ) }$
Of course, this notation can be easily presented in simpler form (fraction), so I've added to analysis the next function - f2(x):
$\displaystyle f(x)= \frac{ c \cdot a_{0} + ( \frac{ c \cdot a_{1}+ \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} }{ a_{1}+ c \cdot \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} } ) }{ a_{0} + c \cdot ( \frac{ c \cdot a_{1}+ \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} }{ a_{1}+ c \cdot \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} } ) }$
So after that, I started to wonder, how this "general" formula of f(x) will looks like when, I will increment number of function - f2(x), f3(x),..., fn(x). Unfortunately, I don't see any pattern related to changes of parameters due to adding next functions, so I'm not able to define general function f(x) for n-number of functions.

I hope that this is way more best explanation of my problem.

Last edited by skipjack; May 26th, 2019 at 03:55 PM.
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May 24th, 2019, 02:41 PM   #4
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Ok, so I'll try to start from the beginning

In last time, I was thinking about what will happen If I take simple function, and replace some parameter in it by using some other or the same function. So in the first step, I defined function f(x):
$\displaystyle f(x)= \frac{c \cdot a_{0}+ b }{ a_{0}+ c \cdot b} $
In this formula I've used parameter b, and replaced it by using function f1(x):
$\displaystyle f_{1}(x)= \frac{c \cdot a_{1}+ f_{2}(x) }{ a_{1}+ c \cdot f_{2}(x)} $

after that, my idea was to find a new, general form of function f(x), by substituting f1(x) to the previous f(x), i.e.:
$\displaystyle f(x)= \frac{ c \cdot a_{0} + ( \frac{ c \cdot a_{1}+ f_{2}(x) }{ a_{1}+ c \cdot f_{2}(x)} ) }{ a_{0} + c \cdot ( \frac{ c \cdot a_{1}+ f_{2}(x) }{ a_{1}+ c \cdot f_{2}(x)} ) }$
Of course, this notation can be easily presented in simpler form (fraction), so I've added to analysis the next function - f2(x):
$\displaystyle f(x)= \frac{ c \cdot a_{0} + ( \frac{ c \cdot a_{1}+ \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} }{ a_{1}+ c \cdot \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} } ) }{ a_{0} + c \cdot ( \frac{ c \cdot a_{1}+ \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} }{ a_{1}+ c \cdot \frac{ c \cdot a_{2}+ f_{3}(x) }{ a_{2}+ c \cdot f_{3}(x)} } ) }$
So after that, I started to wonder, how this "general" formula of f(x) will looks like when, I will increment number of function - f2(x), f3(x),..., fn(x). Unfortunately, I don't see any pattern related to changes of parameters due to adding next functions, so I'm not able to define general function f(x) for n-number of functions.

I hope that this is way much better explanation of my problem.

Last edited by skipjack; May 26th, 2019 at 03:57 PM.
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May 26th, 2019, 02:41 PM   #5
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Mathman,

I've done mistake in this functions - I should correlate all functions with changing of c parameter, NOT x (so we should have F(c), f1(c), and so on). Furthermore, I'm wondering whether I can use the linear fractional transformation theory to solve this problem. It can work ?

Regards,

P.S.
One more thing:
to be clear - I want to present such function $\displaystyle f(c) = \frac{c \cdot a_{0}+\frac{c \cdot a_{1}+\frac{c \cdot a_{2}+f_{3}(c)}{ a_{2}+ c \cdot f_{3}(c)}}{ a_{1}+ c \cdot \frac{c \cdot a_{2}+f_{3}(c)}{ a_{2}+ c \cdot f_{3}(c)}}}{ a_{0}+ c \cdot \frac{c \cdot a_{1}+\frac{c \cdot a_{2}+f_{3}(c)}{ a_{2}+ c \cdot f_{3}(c)}}{ a_{1}+ c \cdot \frac{c \cdot a_{2}+f_{3}(c)}{ a_{2}+ c \cdot f_{3}(c)}}}$


in much more elegant, rational function form - I've problem with that.

Last edited by skipjack; May 26th, 2019 at 03:58 PM.
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