Editing SSC/Structure/OtherAnalyticRamblings (section)

=====Second Guess=====
Let's try again, keeping the same values of the <math>~b_0</math> and <math>~b_2</math> &#8212; that is,
<div align="center">
<math>~(b_0 + b_2x^2) = (2-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~b_0 = 2</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~b_2 = -1</math>
</div>

&#8212; but leaving the values of <math>~a_0</math> and <math>~a_2</math> unspecified.  In this case, the LAWE becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (2 - x^2) + 2x^2(2n a_2 - m a_0)  - 2x^4 (na_2 + m a_2) \biggr](5-3x^2)(a_0 + a_2x^2) 
-\sigma^2 (a_0 + a_2x^2)^2 (2 - x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ n a_2(2 - x^2) -m  (a_0 + a_2x^2)  + 4(2n a_2 - m a_0)  - 4(na_2 + m a_2)x^2 - 4n m a_2 x^2
\biggr](1-x^2)(2-x^2)(a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[2n(n-1) a_2^2(2 - x^2)^2   + 2m(m-1) (a_0 + a_2x^2)^2 \biggr]x^2 (1-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ]  - x^4 [(2na_2 + 2ma_2 ) + a_2\alpha  ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4] 
-\sigma^2 [ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ] (a_0+a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ ( 5m a_0 - 10n a_2)  +  (4n m a_2 + 5na_2 + 5m a_2)x^2 \biggr] (1-x^2)[ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ] + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]x^2 + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]x^4    \biggr\} x^2 (1-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ]  - x^4 [(2na_2 + 2ma_2 ) + a_2\alpha  ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4] 
</math>
  </td> 
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\sigma^2\biggl\{ 2a_0^2 + [2a_0a_2 + a_0(2a_2-a_0)]x^2 +[a_2(2a_2-a_0) -a_0a_2]x^4 - a_2^2 x^6  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{ [ ( 5m a_0 - 10n a_2)  ] + [(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ]x^2 - [ 4n m a_2 + 5na_2 + 5m a_2  ]x^4 \biggr\}
[ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ]
</math>
  </td> 
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]x^2 + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]x^4 + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]x^6    \biggr\} (1-x^2) \, .
</math>
  </td> 
</tr>

</table>
</div>


So, the coefficients of each even power of <math>~x^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~
10a_0^2 \alpha - 2a_0^2\sigma^2 + 2a_0[ ( 5m a_0 - 10n a_2)  ]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~5a_0[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ] + \alpha [2a_0 ](5a_2-3a_0)-\sigma^2[2a_0a_2 + a_0(2a_2-a_0)]
</math><p>
<math>~+ 2a_0[(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ] + (2a_2-a_0)[ ( 5m a_0 - 10n a_2)  ] - [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
</math></p>
 </td>
</tr>

<tr>
  <td align="right"><math>~x^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ - 5a_0[(2na_2 + 2ma_2 ) + a_2\alpha  ] + (5a_2-3a_0)[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ] - 6a_0 a_2\alpha -\sigma^2[a_2(2a_2-a_0) -a_0a_2] 
</math><p>
<math>~- 2a_0[ 4n m a_2 + 5na_2 + 5m a_2  ] + (2a_2-a_0)[(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ] - a_2(5ma_0 - 10na_2)
</math></p><p>
<math>~-[ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ] + [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ -3a_2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ] - (5a_2-3a_0)[(2na_2 + 2ma_2 ) + a_2\alpha  ] +\sigma^2 a_2^2
</math><p>
<math>~- (2a_2-a_0)[ 4n m a_2 + 5na_2 + 5m a_2  ] -  a_2[(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ] 
</math></p><p>
<math>~- [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ] + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 3a_2 a_2\alpha + a_2[ 4n m a_2 + 11na_2 + 11m a_2  ] + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]
</math>
  </td>
</tr>
</table>
</div>


After simplification:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~
10a_0^2 \alpha - 2a_0^2\sigma^2 + 10m a_0^2 - 20n a_0a_2
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~\alpha (20a_0a_2-11a_0^2)    -\sigma^2[4a_0a_2 -a_0^2]  
</math><p>
<math>~+ 60na_0a_2 -20na_2^2  + 20m a_0a_2 -25m a_0^2   + 8n m a_0a_2 
- [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
</math></p>
 </td>
</tr>

<tr>
  <td align="right"><math>~x^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \alpha (10a_2^2 - 22a_0a_2+3a_0^2) -\sigma^2 (2a_2^2 -2a_0a_2)
-47n a_0a_2  +  60n a_2^2 - 50ma_0a_2 +  11m a_0^2 + 10m a_2^2-12n m a_0a_2  + 8n m a_2^2
</math><p>
<math>~+ 16n(n-1) a_2^2 - 4m(m-1)a_0 a_2  + 2m(m-1)a_0^2 
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \alpha (-11a_2^2 + 6a_0 a_2) +\sigma^2 a_2^2 -47n a_2^2 + 11na_0a_2+ 22 m a_0a_2 -25ma_2^2   -12n m a_2^2  + 4n m a_0a_2
</math><p>
<math>~-10n(n-1) a_2^2  - 2m(m-1)a_2^2  + 4m(m-1)a_0 a_2 
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \{ 3\alpha + [ 4n m  + 11n + 11m   ] + [ 2n(n-1)   + 2m(m-1) ]\}a_2^2
</math>
  </td>
</tr>
</table>
</div>



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