Discrete mathematics--An easy doubt on the notations of sums

[V9999]

TL;DR

Here, I present a silly question about the notation of sums.

I have a doubt about the notation and alternative ways to represent the terms involved in sums.

Suppose that we have the following multivariable function,

$$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$.

Now, let $\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}$. In the light of the foregoing, is it correct to express $f(x,y)$ as follows

$$f(x,y)=\sum^{m}_{j=0}y^{j}\psi_{j}(x)$$ ?

Thanks in advance!

 

[fresh_42]

TL;DR Summary: Here, I present a silly question about the notation of sums.

I have a doubt about the notation and alternative ways to represent the terms involved in sums.

Suppose that we have the following multivariable function,

$$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$.

Now, let $\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}$. In the light of the foregoing, is it correct to express $f(x,y)$ as follows

$$f(x,y)=\sum^{m}_{j=0}y^{j}\psi_{j}(x)$$ ?

Thanks in advance!

Yes.

I think - not sure, look it up - with Einstein's summation convention you can even write $f(x,y)=y^j\psi_j(x).$

 

[FactChecker]

Now, let $\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}$. In the light of the foregoing, is it correct to express $f(x,y)$ as follows

$$f(x,y)=\sum^{m}_{j=0}y^{j}\psi_{j}(x)$$ ?

Yes, but you might want to be specific in your definition of $\psi_j$ about the legitimate values of $j$. In the original summation, the legitimate values of $j$ are known.

 

[mathman]

TL;DR Summary: Here, I present a silly question about the notation of sums.

I have a doubt about the notation and alternative ways to represent the terms involved in sums.

Suppose that we have the following multivariable function,

$$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$.

Now, let $\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}$. In the light of the foregoing, is it correct to express $f(x,y)$ as follows

$$f(x,y)=\sum^{m}_{j=0}y^{j}\psi_{j}(x)$$ ?

Thanks in advance!

$j-m\le 0$. Inner sum is strange.

 

[Office_Shredder]

I would rather write that as $\psi_{j,m}(x)$ to avoid any ambiguity. Also what mathman said...

 

[V9999]

Yes.

I think - not sure, look it up - with Einstein's summation convention you can even write $f(x,y)=y^j\psi_j(x).$

Hi, fresh_42. I hope you are doing well. Thank you very and very much for your comments.

 

[V9999]

Yes, but you might want to be specific in your definition of $\psi_j$ about the legitimate values of $j$. In the original summation, the legitimate values of $j$ are known.

Hi, FactChecker. I hope you are doing well. Thanks for the great insight and I will take it under consideration.

 

[V9999]

I would rather write that as $\psi_{j,m}(x)$ to avoid any ambiguity. Also what mathman said...

Hi, Office_Shredder. I hope you are doing well. Thanks for the great insight and I will take it under consideration.