On the roots of Bernoulli polynomials
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Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".
QUESTION: Or, does it? If so, what are these curves?
Request: Can someone post the complex plot here?
reference-request cv.complex-variables soft-question polynomials
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
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add a comment |
$begingroup$
Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".
QUESTION: Or, does it? If so, what are these curves?
Request: Can someone post the complex plot here?
reference-request cv.complex-variables soft-question polynomials
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
$endgroup$
5
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One reference is arxiv.org/pdf/math/0703452.pdf.
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– Richard Stanley
Jan 23 at 3:44
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@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43
add a comment |
$begingroup$
Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".
QUESTION: Or, does it? If so, what are these curves?
Request: Can someone post the complex plot here?
reference-request cv.complex-variables soft-question polynomials
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
$endgroup$
Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".
QUESTION: Or, does it? If so, what are these curves?
Request: Can someone post the complex plot here?
reference-request cv.complex-variables soft-question polynomials
reference-request cv.complex-variables soft-question polynomials
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
asked Jan 23 at 3:15
T. AmdeberhanT. Amdeberhan
17.2k229127
17.2k229127
5
$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
Jan 23 at 3:44
$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43
add a comment |
5
$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
Jan 23 at 3:44
$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43
5
5
$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
Jan 23 at 3:44
$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
Jan 23 at 3:44
$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43
$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43
add a comment |
1 Answer
1
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Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.
For the number of real roots, see OEIS sequence A094937 and references there.
EDIT: As requested by Wolfgang, here is a plot of the real roots for even $n$ up to $200$.
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
$endgroup$
$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44
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That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n?
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– Wolfgang
Jan 26 at 15:31
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Very nice! Can share the code, please? Put it here or link to github or like that...
$endgroup$
– Alexander Chervov
Jan 27 at 14:41
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plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box);
$endgroup$
– Robert Israel
Jan 27 at 16:45
add a comment |
Your Answer
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1 Answer
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1 Answer
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active
oldest
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active
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votes
$begingroup$
Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.
For the number of real roots, see OEIS sequence A094937 and references there.
EDIT: As requested by Wolfgang, here is a plot of the real roots for even $n$ up to $200$.
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
$endgroup$
$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44
$begingroup$
That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n?
$endgroup$
– Wolfgang
Jan 26 at 15:31
$begingroup$
Very nice! Can share the code, please? Put it here or link to github or like that...
$endgroup$
– Alexander Chervov
Jan 27 at 14:41
$begingroup$
plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box);
$endgroup$
– Robert Israel
Jan 27 at 16:45
add a comment |
$begingroup$
Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.
For the number of real roots, see OEIS sequence A094937 and references there.
EDIT: As requested by Wolfgang, here is a plot of the real roots for even $n$ up to $200$.
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
$endgroup$
$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44
$begingroup$
That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n?
$endgroup$
– Wolfgang
Jan 26 at 15:31
$begingroup$
Very nice! Can share the code, please? Put it here or link to github or like that...
$endgroup$
– Alexander Chervov
Jan 27 at 14:41
$begingroup$
plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box);
$endgroup$
– Robert Israel
Jan 27 at 16:45
add a comment |
$begingroup$
Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.
For the number of real roots, see OEIS sequence A094937 and references there.
EDIT: As requested by Wolfgang, here is a plot of the real roots for even $n$ up to $200$.
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
$endgroup$
Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.
For the number of real roots, see OEIS sequence A094937 and references there.
EDIT: As requested by Wolfgang, here is a plot of the real roots for even $n$ up to $200$.
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
edited Jan 27 at 3:14
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
answered Jan 23 at 3:54
Robert IsraelRobert Israel
41.9k51120
41.9k51120
$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44
$begingroup$
That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n?
$endgroup$
– Wolfgang
Jan 26 at 15:31
$begingroup$
Very nice! Can share the code, please? Put it here or link to github or like that...
$endgroup$
– Alexander Chervov
Jan 27 at 14:41
$begingroup$
plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box);
$endgroup$
– Robert Israel
Jan 27 at 16:45
add a comment |
$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44
$begingroup$
That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n?
$endgroup$
– Wolfgang
Jan 26 at 15:31
$begingroup$
Very nice! Can share the code, please? Put it here or link to github or like that...
$endgroup$
– Alexander Chervov
Jan 27 at 14:41
$begingroup$
plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box);
$endgroup$
– Robert Israel
Jan 27 at 16:45
$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44
$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44
$begingroup$
That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n?
$endgroup$
– Wolfgang
Jan 26 at 15:31
$begingroup$
That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n?
$endgroup$
– Wolfgang
Jan 26 at 15:31
$begingroup$
Very nice! Can share the code, please? Put it here or link to github or like that...
$endgroup$
– Alexander Chervov
Jan 27 at 14:41
$begingroup$
Very nice! Can share the code, please? Put it here or link to github or like that...
$endgroup$
– Alexander Chervov
Jan 27 at 14:41
$begingroup$
plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box);
$endgroup$
– Robert Israel
Jan 27 at 16:45
$begingroup$
plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box);
$endgroup$
– Robert Israel
Jan 27 at 16:45
add a comment |
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5
$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
Jan 23 at 3:44
$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43