Sometimes I like to find patterns in certain functions, for example, repeated Sigma (Summation) notation.(adsbygoogle = window.adsbygoogle || []).push({});

But what if I wanted to do an arbitrary number of nested summations? Or something similar with other functions? Is there a compressed way of writing this? For example:

[tex] \sum_{k_5=1}^{1} \left (k_5 \sum_{k_4=2}^{4} \left (k_4 \sum_{k_3=4}^{16} \left (k_3 \sum_{k_2=8}^{64} \left (k_2 \sum_{k_1=16}^{256}\left ( k_1f(x) \right ) \right ) \right ) \right ) \right ) [/tex]

I have come up with a large operator for this, very similar to the Sigma notation, but using a pair of extra square brackets and Omega for the operator.

[tex] \Omega_{l=1}^{5}\left [ \sum_{k_l=2^{5-l}}^{4^{5-l}} (k_lx) \right ]_x\left ( f(x) \right ) [/tex]

It looks similar to this, but the Omega would be larger and the subscript/superscript parts would be on the bottom/top of the Omega respectively.

This notation means:

This could be used with any function, even a single Sigma or Pi large operation or just 1+2!

- Whatever is in the round brackets is the base.
- The subscript pronumeral next to the square brackets denotes what is being replaced inside the square brackets by the base.
- The function inside the square brackets is executed like any other large operator, then the value of l rises from 1 to 2 and the x in the square brackets is then replaced by what you have now.
- This is then repeated until you have executed the 5th function.

Keep in mind that this is a sample of what it could do, even though there may be a much simpler way of writing it, but there are heaps of other functions that this may be useful for (compression-wise, at least, or for finding patterns).

[tex] \Omega_{l=1}^{n} \left [ f_l(x) \right ]_x \left ( x \right ) = f_n(f_{n-1}(\cdots f_2(f_1(x))\cdots))[/tex]

Or...

[tex] \Omega_{l=1}^{1} \left [ x+2 \right ]_x \left ( 1 \right ) = 3[/tex]

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# Large Operator for Composite Functions

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