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The Euler–Frobenius polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202101.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202102.png" /> are characterized by the Frobenius reciprocal identity ([[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]])
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If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202103.png" /></td> </tr></table>
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Out of 48 formulas, 48 were replaced by TEX code.-->
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202104.png" /> is invariant under the reflection
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The Euler–Frobenius polynomials $p _ { m } ( x )$ of degree $m - 1 \geq 0$ are characterized by the Frobenius reciprocal identity ([[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202105.png" /></td> </tr></table>
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\begin{equation*} x ^ { m - 1 } p _ { m } \left( \frac { 1 } { x } \right) = p _ { m } ( x ). \end{equation*}
  
of the indeterminate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202106.png" />. The best way to implement an invariance of this kind is to look for an appropriate space with which the Euler–Frobenius polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202107.png" /> are attached in a spectral geometric way.
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Thus, $p _ { m } ( x )$ is invariant under the reflection
  
So, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202108.png" /> denote a symplectic vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e1202109.png" /> (cf. also [[Symplectic space|Symplectic space]]). Then the [[Characteristic polynomial|characteristic polynomial]] of each symplectic automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021010.png" /> is an Euler–Frobenius polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021011.png" /> of odd degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021012.png" />.
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\begin{equation*} x \rightarrow \frac { 1 } { x } \end{equation*}
  
The proof follows from the fact that the determinant of each symplectic automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021014.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021015.png" />, so that there is a natural imbedding
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of the indeterminate $x$. The best way to implement an invariance of this kind is to look for an appropriate space with which the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ are attached in a spectral geometric way.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021016.png" /></td> </tr></table>
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So, let $E$ denote a symplectic vector space of dimension $m = 2 n$ (cf. also [[Symplectic space|Symplectic space]]). Then the [[Characteristic polynomial|characteristic polynomial]] of each symplectic automorphism of $E$ is an Euler–Frobenius polynomial $p _ { m } ( x )$ of odd degree $m - 1$.
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021017.png" /> preserves the symplectic volume spanned by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021018.png" /> vectors of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021019.png" />.
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The proof follows from the fact that the determinant of each symplectic automorphism $\sigma$ of $E$ equals $1$, so that there is a natural imbedding
  
A consequence is that each eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021020.png" /> of a symplectic endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021022.png" /> having multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021023.png" /> gives rise to a reciprocal eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021024.png" /> of the same multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021025.png" />.
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\begin{equation*} \operatorname { Sp } ( E ) \hookrightarrow \operatorname { SL } ( E ). \end{equation*}
 +
 
 +
Thus, $\sigma \in \operatorname{Sp} ( E )$ preserves the symplectic volume spanned by $m$ vectors of the vector space $E$.
 +
 
 +
A consequence is that each eigenvalue $\lambda \neq 0$ of a symplectic endomorphism $\sigma$ of $E$ having multiplicity $k$ gives rise to a reciprocal eigenvalue $1 / \lambda$ of the same multiplicity $k$.
  
 
In view of the self-reciprocal eigenvalue
 
In view of the self-reciprocal eigenvalue
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021026.png" /></td> </tr></table>
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\begin{equation*} \lambda _ { 0 } = - 1 \end{equation*}
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021027.png" /> for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021028.png" />, of course, spectral theory suggests a complex contour integral representation of the Euler–Frobenius polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021029.png" />, as follows.
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of $\sigma : E \rightarrow E$ for even $m = 2 n$, of course, spectral theory suggests a complex contour integral representation of the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$, as follows.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021030.png" /> denote a complex number such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021031.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021032.png" /> denote a path in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021033.png" /> which forms the boundary of a closed vertical strip in the open right or left half-plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021034.png" /> according as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021035.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021036.png" />, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021037.png" /> be oriented so that its topological index satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021038.png" />. Then, for each integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021039.png" />, the complex contour integral representation
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Let $z \neq 0$ denote a complex number such that $| z | \neq 1$. Let $P$ denote a path in the complex plane $\mathbf{C}$ which forms the boundary of a closed vertical strip in the open right or left half-plane of $\mathbf{C}$ according as $| z | > 1$ or $0 < | z | < 1$, respectively. Let $P \hookrightarrow \mathbf{C}$ be oriented so that its topological index satisfies $\operatorname { ind } _ { P } ( \operatorname { log } | z | ) = 1$. Then, for each integer $m \geq 1$, the complex contour integral representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021040.png" /></td> </tr></table>
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\begin{equation*} p _ { m } ( z ) = \frac { ( z - 1 ) ^ { m + 1 } } { z } \frac { m ! } { 2 \pi i } \int _ { P } \frac { e ^ { w } } { ( e ^ { w } - z ) w ^ { m + 1 } } d w \end{equation*}
  
 
holds.
 
holds.
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The proof follows from the expansion
 
The proof follows from the expansion
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021041.png" /></td> </tr></table>
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\begin{equation*} p _ { m } ( z ) = m ! \sum _ { 0 \leq n \leq m - 1 } b _ { m } ( n + 1 ) z ^ { n } , \quad z \in \mathbf{C}, \end{equation*}
  
with strictly positive integer coefficients, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021042.png" /> denote the basis spline functions (cf. [[#References|[a3]]] and also [[Spline|Spline]]).
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with strictly positive integer coefficients, where $( b _ { m } ) _ { m \geq 0 }$ denote the basis spline functions (cf. [[#References|[a3]]] and also [[Spline|Spline]]).
  
 
A consequence is that the Euler–Frobenius polynomials provide the coefficients of the local power series expansion of the function
 
A consequence is that the Euler–Frobenius polynomials provide the coefficients of the local power series expansion of the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021043.png" /></td> </tr></table>
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\begin{equation*} w \rightarrow \frac { ( z - 1 ) e ^ { w } } { z ( z - e ^ { w } ) } , \quad z \in \mathbf{C}, \end{equation*}
  
which is meromorphic on the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021044.png" />.
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which is meromorphic on the complex plane $\mathbf{C}$.
  
The Euler–Frobenius polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021045.png" /> satisfy the three-term recurrence relation
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The Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ satisfy the three-term recurrence relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021046.png" /></td> </tr></table>
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\begin{equation*} p _ { m  + 1} ( x ) = ( m x + 1 ) p _ { m } ( x ) - x ( x - 1 ) p _ { m } ^ { \prime } ( x ) , \quad m \geq 1. \end{equation*}
  
A direct proof follows from the complex contour integral representations of the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021047.png" />, which can be derived from the complex contour integral representation given above for the Euler–Frobenius polynomials.
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A direct proof follows from the complex contour integral representations of the derivatives $( p _ { m } ^ { \prime } ( x ) ) _ { m \geq 1 }$, which can be derived from the complex contour integral representation given above for the Euler–Frobenius polynomials.
  
The preceding recurrence relation opens a simple way to calculate the coefficients of the Euler–Frobenius polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021048.png" /> ([[#References|[a1]]], [[#References|[a3]]]).
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The preceding recurrence relation opens a simple way to calculate the coefficients of the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ ([[#References|[a1]]], [[#References|[a3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Euler,  "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum" , Acad. Imper. Sci. Petropolitanæ  (1775)  (Opera Omnis Ser. I (Opera Math.), Vol. X, Teubner, 1913)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.G. Frobenius,  "Über die Bernoullischen Zahlen und die Eulerschen Polynome"  ''Sitzungsber. K. Preuss. Akad. Wissenschaft. Berlin''  (1910)  pp. 809–847  (Gesammelte Abh. Vol. III, pp. 440-478, Springer 1968)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Schempp,  "Complex contour integral representation of cardinal spline functions" , ''Contemp. Math.'' , '''7''' , Amer. Math. Soc.  (1982)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  L. Euler,  "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum" , Acad. Imper. Sci. Petropolitanæ  (1775)  (Opera Omnis Ser. I (Opera Math.), Vol. X, Teubner, 1913)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  F.G. Frobenius,  "Über die Bernoullischen Zahlen und die Eulerschen Polynome"  ''Sitzungsber. K. Preuss. Akad. Wissenschaft. Berlin''  (1910)  pp. 809–847  (Gesammelte Abh. Vol. III, pp. 440-478, Springer 1968)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  W. Schempp,  "Complex contour integral representation of cardinal spline functions" , ''Contemp. Math.'' , '''7''' , Amer. Math. Soc.  (1982)</td></tr>
 +
</table>

Latest revision as of 07:27, 25 January 2024

The Euler–Frobenius polynomials $p _ { m } ( x )$ of degree $m - 1 \geq 0$ are characterized by the Frobenius reciprocal identity ([a1], [a2], [a3])

\begin{equation*} x ^ { m - 1 } p _ { m } \left( \frac { 1 } { x } \right) = p _ { m } ( x ). \end{equation*}

Thus, $p _ { m } ( x )$ is invariant under the reflection

\begin{equation*} x \rightarrow \frac { 1 } { x } \end{equation*}

of the indeterminate $x$. The best way to implement an invariance of this kind is to look for an appropriate space with which the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ are attached in a spectral geometric way.

So, let $E$ denote a symplectic vector space of dimension $m = 2 n$ (cf. also Symplectic space). Then the characteristic polynomial of each symplectic automorphism of $E$ is an Euler–Frobenius polynomial $p _ { m } ( x )$ of odd degree $m - 1$.

The proof follows from the fact that the determinant of each symplectic automorphism $\sigma$ of $E$ equals $1$, so that there is a natural imbedding

\begin{equation*} \operatorname { Sp } ( E ) \hookrightarrow \operatorname { SL } ( E ). \end{equation*}

Thus, $\sigma \in \operatorname{Sp} ( E )$ preserves the symplectic volume spanned by $m$ vectors of the vector space $E$.

A consequence is that each eigenvalue $\lambda \neq 0$ of a symplectic endomorphism $\sigma$ of $E$ having multiplicity $k$ gives rise to a reciprocal eigenvalue $1 / \lambda$ of the same multiplicity $k$.

In view of the self-reciprocal eigenvalue

\begin{equation*} \lambda _ { 0 } = - 1 \end{equation*}

of $\sigma : E \rightarrow E$ for even $m = 2 n$, of course, spectral theory suggests a complex contour integral representation of the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$, as follows.

Let $z \neq 0$ denote a complex number such that $| z | \neq 1$. Let $P$ denote a path in the complex plane $\mathbf{C}$ which forms the boundary of a closed vertical strip in the open right or left half-plane of $\mathbf{C}$ according as $| z | > 1$ or $0 < | z | < 1$, respectively. Let $P \hookrightarrow \mathbf{C}$ be oriented so that its topological index satisfies $\operatorname { ind } _ { P } ( \operatorname { log } | z | ) = 1$. Then, for each integer $m \geq 1$, the complex contour integral representation

\begin{equation*} p _ { m } ( z ) = \frac { ( z - 1 ) ^ { m + 1 } } { z } \frac { m ! } { 2 \pi i } \int _ { P } \frac { e ^ { w } } { ( e ^ { w } - z ) w ^ { m + 1 } } d w \end{equation*}

holds.

The proof follows from the expansion

\begin{equation*} p _ { m } ( z ) = m ! \sum _ { 0 \leq n \leq m - 1 } b _ { m } ( n + 1 ) z ^ { n } , \quad z \in \mathbf{C}, \end{equation*}

with strictly positive integer coefficients, where $( b _ { m } ) _ { m \geq 0 }$ denote the basis spline functions (cf. [a3] and also Spline).

A consequence is that the Euler–Frobenius polynomials provide the coefficients of the local power series expansion of the function

\begin{equation*} w \rightarrow \frac { ( z - 1 ) e ^ { w } } { z ( z - e ^ { w } ) } , \quad z \in \mathbf{C}, \end{equation*}

which is meromorphic on the complex plane $\mathbf{C}$.

The Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ satisfy the three-term recurrence relation

\begin{equation*} p _ { m + 1} ( x ) = ( m x + 1 ) p _ { m } ( x ) - x ( x - 1 ) p _ { m } ^ { \prime } ( x ) , \quad m \geq 1. \end{equation*}

A direct proof follows from the complex contour integral representations of the derivatives $( p _ { m } ^ { \prime } ( x ) ) _ { m \geq 1 }$, which can be derived from the complex contour integral representation given above for the Euler–Frobenius polynomials.

The preceding recurrence relation opens a simple way to calculate the coefficients of the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ ([a1], [a3]).

References

[a1] L. Euler, "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum" , Acad. Imper. Sci. Petropolitanæ (1775) (Opera Omnis Ser. I (Opera Math.), Vol. X, Teubner, 1913)
[a2] F.G. Frobenius, "Über die Bernoullischen Zahlen und die Eulerschen Polynome" Sitzungsber. K. Preuss. Akad. Wissenschaft. Berlin (1910) pp. 809–847 (Gesammelte Abh. Vol. III, pp. 440-478, Springer 1968)
[a3] W. Schempp, "Complex contour integral representation of cardinal spline functions" , Contemp. Math. , 7 , Amer. Math. Soc. (1982)
How to Cite This Entry:
Euler-Frobenius polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler-Frobenius_polynomials&oldid=17228
This article was adapted from an original article by Walter Schempp (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article