Mixing
This transform gives the coefficients of the preimage of a polynomial isomorphism; what is its name?
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The map $$alpha_0 + alpha_1 x + cdots + alpha_n x^n mapsto alpha_0 + alpha_1 (x + 1) + cdots + alpha_n (x + 1)^n$$ defines an isomorphism from $F_n[x] = P in F[x] mid textrmdeg P leq n$ into itself, where $F$ is some field.
To see that the map is surjective, given some polynomial $beta_0 + beta_1 x + cdots + beta_n x^n$, we must determine coefficients $alpha_0, dots, alpha_n$ such that $$alpha_0 + alpha_1 (x + 1) + alpha_2 (x + 1)^2 + cdots + alpha_n (x + 1)^n = beta_0 + beta_1 x + cdots + beta_n x^n.$$
This is not very hard to do; by applying the binomial theorem to the left-hand side and equating the resulting coefficients, we get the recurrence $$alpha_n - r = beta_n - r - sum_k = n - r + 1^n k choose n - r alpha_k, qquad r = 0, 1, dots, n.$$
Computing backwards from $r = n$ to $r = 0$, we can uniquely transform the $beta_k$'s into the $alpha_k$'s.
Is there a name for this transformation of $beta_k_k = 0^n$ into $alpha_k_0^n$?
sequences-and-series polynomials summation terminology
edited Jul 23 at 4:14
joriki
164k10180328
asked Jul 23 at 4:05
rwbogl
716415
add a comment |Â
up vote
0
down vote
favorite
The map $$alpha_0 + alpha_1 x + cdots + alpha_n x^n mapsto alpha_0 + alpha_1 (x + 1) + cdots + alpha_n (x + 1)^n$$ defines an isomorphism from $F_n[x] = P in F[x] mid textrmdeg P leq n$ into itself, where $F$ is some field.
To see that the map is surjective, given some polynomial $beta_0 + beta_1 x + cdots + beta_n x^n$, we must determine coefficients $alpha_0, dots, alpha_n$ such that $$alpha_0 + alpha_1 (x + 1) + alpha_2 (x + 1)^2 + cdots + alpha_n (x + 1)^n = beta_0 + beta_1 x + cdots + beta_n x^n.$$
This is not very hard to do; by applying the binomial theorem to the left-hand side and equating the resulting coefficients, we get the recurrence $$alpha_n - r = beta_n - r - sum_k = n - r + 1^n k choose n - r alpha_k, qquad r = 0, 1, dots, n.$$
Computing backwards from $r = n$ to $r = 0$, we can uniquely transform the $beta_k$'s into the $alpha_k$'s.
Is there a name for this transformation of $beta_k_k = 0^n$ into $alpha_k_0^n$?
sequences-and-series polynomials summation terminology
edited Jul 23 at 4:14
joriki
164k10180328
asked Jul 23 at 4:05
rwbogl
716415
1
Hmm. Were we to choose $F = mathbbC$ or $F = mathbbR$ what you are asking for is sometimes referred to merely as a change of variables in the power series, and thus I haven't ever needed a name before. Not an answer at all, simply my experiences with this
– Brevan Ellefsen
Jul 23 at 4:15
I might be trying to give a fancy name to something pretty simple - I appreciate the heads up!
– rwbogl
Jul 23 at 4:34
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The map $$alpha_0 + alpha_1 x + cdots + alpha_n x^n mapsto alpha_0 + alpha_1 (x + 1) + cdots + alpha_n (x + 1)^n$$ defines an isomorphism from $F_n[x] = P in F[x] mid textrmdeg P leq n$ into itself, where $F$ is some field.
To see that the map is surjective, given some polynomial $beta_0 + beta_1 x + cdots + beta_n x^n$, we must determine coefficients $alpha_0, dots, alpha_n$ such that $$alpha_0 + alpha_1 (x + 1) + alpha_2 (x + 1)^2 + cdots + alpha_n (x + 1)^n = beta_0 + beta_1 x + cdots + beta_n x^n.$$
This is not very hard to do; by applying the binomial theorem to the left-hand side and equating the resulting coefficients, we get the recurrence $$alpha_n - r = beta_n - r - sum_k = n - r + 1^n k choose n - r alpha_k, qquad r = 0, 1, dots, n.$$
Computing backwards from $r = n$ to $r = 0$, we can uniquely transform the $beta_k$'s into the $alpha_k$'s.
Is there a name for this transformation of $beta_k_k = 0^n$ into $alpha_k_0^n$?
sequences-and-series polynomials summation terminology
edited Jul 23 at 4:14
joriki
164k10180328
asked Jul 23 at 4:05
rwbogl
716415
The map $$alpha_0 + alpha_1 x + cdots + alpha_n x^n mapsto alpha_0 + alpha_1 (x + 1) + cdots + alpha_n (x + 1)^n$$ defines an isomorphism from $F_n[x] = P in F[x] mid textrmdeg P leq n$ into itself, where $F$ is some field.
To see that the map is surjective, given some polynomial $beta_0 + beta_1 x + cdots + beta_n x^n$, we must determine coefficients $alpha_0, dots, alpha_n$ such that $$alpha_0 + alpha_1 (x + 1) + alpha_2 (x + 1)^2 + cdots + alpha_n (x + 1)^n = beta_0 + beta_1 x + cdots + beta_n x^n.$$
This is not very hard to do; by applying the binomial theorem to the left-hand side and equating the resulting coefficients, we get the recurrence $$alpha_n - r = beta_n - r - sum_k = n - r + 1^n k choose n - r alpha_k, qquad r = 0, 1, dots, n.$$
Computing backwards from $r = n$ to $r = 0$, we can uniquely transform the $beta_k$'s into the $alpha_k$'s.
Is there a name for this transformation of $beta_k_k = 0^n$ into $alpha_k_0^n$?
sequences-and-series polynomials summation terminology
edited Jul 23 at 4:14
joriki
164k10180328
asked Jul 23 at 4:05
rwbogl
716415
edited Jul 23 at 4:14
joriki
164k10180328
edited Jul 23 at 4:14
joriki
164k10180328
edited Jul 23 at 4:14
joriki
164k10180328
164k10180328
asked Jul 23 at 4:05
rwbogl
716415
asked Jul 23 at 4:05
rwbogl
716415
asked Jul 23 at 4:05
rwbogl
716415
716415
1
Hmm. Were we to choose $F = mathbbC$ or $F = mathbbR$ what you are asking for is sometimes referred to merely as a change of variables in the power series, and thus I haven't ever needed a name before. Not an answer at all, simply my experiences with this
– Brevan Ellefsen
Jul 23 at 4:15
I might be trying to give a fancy name to something pretty simple - I appreciate the heads up!
– rwbogl
Jul 23 at 4:34
add a comment |Â
1
Hmm. Were we to choose $F = mathbbC$ or $F = mathbbR$ what you are asking for is sometimes referred to merely as a change of variables in the power series, and thus I haven't ever needed a name before. Not an answer at all, simply my experiences with this
– Brevan Ellefsen
Jul 23 at 4:15
I might be trying to give a fancy name to something pretty simple - I appreciate the heads up!
– rwbogl
Jul 23 at 4:34
1
1
Hmm. Were we to choose $F = mathbbC$ or $F = mathbbR$ what you are asking for is sometimes referred to merely as a change of variables in the power series, and thus I haven't ever needed a name before. Not an answer at all, simply my experiences with this
– Brevan Ellefsen
Jul 23 at 4:15
Hmm. Were we to choose $F = mathbbC$ or $F = mathbbR$ what you are asking for is sometimes referred to merely as a change of variables in the power series, and thus I haven't ever needed a name before. Not an answer at all, simply my experiences with this
– Brevan Ellefsen
Jul 23 at 4:15
I might be trying to give a fancy name to something pretty simple - I appreciate the heads up!
– rwbogl
Jul 23 at 4:34
I might be trying to give a fancy name to something pretty simple - I appreciate the heads up!
– rwbogl
Jul 23 at 4:34
add a comment |Â
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1
Hmm. Were we to choose $F = mathbbC$ or $F = mathbbR$ what you are asking for is sometimes referred to merely as a change of variables in the power series, and thus I haven't ever needed a name before. Not an answer at all, simply my experiences with this
– Brevan Ellefsen
Jul 23 at 4:15
I might be trying to give a fancy name to something pretty simple - I appreciate the heads up!
– rwbogl
Jul 23 at 4:34