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How may I compute series expansions of expressions with 2 summations signs?

Posted 1 year ago

I have a few questions about series expansions of a particular and difficult type of polynomial written in terms of two summations signs.

It should be remarked that I have also read Mathematica's documentation and previous posts on the Wolfram community. However, those did not help me with the questions to be presented. Based on the great interest in series expansion in one or more variables, I believe these questions and their possible answers may be of great interest to those to subscribe to the Wolfram community.

With that said, suppose that we have the following expression,

$$\sum _{j=0}^n \sum _{i=0}^j (K y+\sigma )^iy^{j-i} a_{i,j-i},$$

in which $n$ is the maximum degree of algebraic expression above. Based on the above, I ask:

  1. Is there another way to perform a second order Taylor series expansion of the expression above using Mathematica?

It is critical to recognize that I have tried to use Mathematica to obtain the Taylor series expansion of the expression above. Indeed, it is possible to perform derivatives of that polynomial expression using Mathematica, as you may see in my notebook attached below. However, when I try to compute series expansion, I receive

Series[Sum[y^(j-i) a[i,j-i](K*y+b)^i,{j,0,n},{i,0,j}],{y,0,2}]

as an output.

Second, how may I determine the maximum order of the resultant expression from the series expansion?

Attachments:

What result do you expect for the example given?

POSTED BY: Daniel Lichtblau
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