View source for Euler-Frobenius polynomials

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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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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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.

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" />.

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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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" />.

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" />.

In view of the self-reciprocal eigenvalue

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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.

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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holds.

The proof follows from the expansion

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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]]).

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

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which is meromorphic on the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120210/e12021044.png" />.

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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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.

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]]]).

====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>