3
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Let's define

$u=t+1/2$

$v=t-1/2$

Now, I have vectors in the basis $e_n=u^n-v^n$, I want to apply functions to such expressions, but as a result get expressions in the same basis again.

For instance, $w=\frac{u^2-v^2}2$, $w^2=\frac{1}{3} \left(u^3-v^3\right)-\frac{u-v}{12}$, so $(0,1/2,0,0...)^2=(1/12,0,1/3,0,0,...)$. Can operations on such vectors be automatized?

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  • 2
    $\begingroup$ Please add the code you have tried and the difficulties you encountered. $\endgroup$
    – bbgodfrey
    Dec 16 '21 at 12:27
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u = t + 1/2;
v = t - 1/2;
e[n_] := u^n - v^n
eq[k_, l_] := Sum[a[i] e[i], {i, 0, k + l}];
res[2, 2] = SolveAlways[e[2]^2 == eq[2, 2], t] // First
Clear[u, v]
eq[2, 2] /. res[2, 2]
(*{{a[1] -> -(1/3), a[3] -> 4/3, a[4] -> 0, a[2] -> 0}}*)

$$(u^2 - v^2)^2=\frac{4}{3} \left(u^3-v^3\right)+\frac{v-u}{3}$$

u = t + 1/2;
v = t - 1/2;
e[n_] := u^n - v^n
eq[k_, l_] := Sum[a[i] e[i], {i, 0, k + l}];
res[3, 5] = 
 SolveAlways[e[3]^5 == Sum[a[i] e[i], {i, 0, 15}], t] // First
Clear[u, v]
eq[3, 5] /. res[3, 5]

(*{{a[14] -> 0, a[2] -> 0, a[4] -> 0, a[6] -> 0, a[8] -> 0, a[10] -> 0, 
  a[12] -> 0, a[15] -> 0, a[1] -> -(1184/77), a[3] -> 101, 
  a[5] -> -198, a[7] -> 1269/7, a[9] -> -90, a[11] -> 243/11, 
  a[13] -> 0}}*)

$$(u^3 - v^3)^5=\frac{1269}{7} \left(u^7-v^7\right)-198 \left(u^5-v^5\right)+101 \left(u^3-v^3\right)-\frac{1184}{77} (u-v)$$

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  • $\begingroup$ The basis is infinite. I doubt this can handle an infinite series... $\endgroup$
    – Anixx
    Dec 16 '21 at 16:23
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    $\begingroup$ @Anixx Please, update your post with an explicit example of infinite basis that you would like to automatize. $\endgroup$
    – yarchik
    Dec 16 '21 at 16:26

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