Euler-Frobenius polynomials

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)