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

In summary: 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.
  • #1
V9999
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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!
 
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  • #2
V9999 said:
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).##
 
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  • #3
V9999 said:
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.
 
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  • #4
V9999 said:
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.
 
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  • #5
I would rather write that as ##\psi_{j,m}(x)## to avoid any ambiguity. Also what mathman said...
 
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  • #6
fresh_42 said:
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.
 
  • #7
FactChecker said:
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.
 
  • #8
Office_Shredder said:
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.
 

1. What is discrete mathematics?

Discrete mathematics is a branch of mathematics that deals with discrete or countable objects, as opposed to continuous objects. It involves the study of structures that are discrete in nature, such as integers, graphs, and logical statements.

2. What are the notations used in sums in discrete mathematics?

The two most commonly used notations in sums are sigma notation (∑) and product notation (∏). Sigma notation is used for sums of a sequence of terms, while product notation is used for products of a sequence of terms.

3. How do you read sigma notation?

Sigma notation is read as "the sum of" or "the summation of". The variable below the sigma symbol represents the index or the starting point of the sum, while the number above the sigma symbol represents the ending point of the sum. The expression after the sigma symbol represents the term to be summed.

4. What is the purpose of using sigma notation in discrete mathematics?

Sigma notation is used to compactly represent and evaluate sums of a large number of terms. It also allows for easier manipulation and calculation of sums, making it a useful tool in various areas of discrete mathematics such as combinatorics, number theory, and probability.

5. Can you provide an example of using product notation in discrete mathematics?

Yes, an example of using product notation is the factorial function, which is denoted by n!. It represents the product of all positive integers from 1 to n. For example, 5! = 1 x 2 x 3 x 4 x 5 = 120.

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