Thus, is invariant under the reflection
of the indeterminate . The best way to implement an invariance of this kind is to look for an appropriate space with which the Euler–Frobenius polynomials are attached in a spectral geometric way.
So, let denote a symplectic vector space of dimension (cf. also Symplectic space). Then the characteristic polynomial of each symplectic automorphism of is an Euler–Frobenius polynomial of odd degree .
The proof follows from the fact that the determinant of each symplectic automorphism of equals , so that there is a natural imbedding
Thus, preserves the symplectic volume spanned by vectors of the vector space .
A consequence is that each eigenvalue of a symplectic endomorphism of having multiplicity gives rise to a reciprocal eigenvalue of the same multiplicity .
In view of the self-reciprocal eigenvalue
of for even , of course, spectral theory suggests a complex contour integral representation of the Euler–Frobenius polynomials , as follows.
Let denote a complex number such that . Let denote a path in the complex plane which forms the boundary of a closed vertical strip in the open right or left half-plane of according as or , respectively. Let be oriented so that its topological index satisfies . Then, for each integer , the complex contour integral representation
The proof follows from the expansion
A consequence is that the Euler–Frobenius polynomials provide the coefficients of the local power series expansion of the function
which is meromorphic on the complex plane .
The Euler–Frobenius polynomials satisfy the three-term recurrence relation
A direct proof follows from the complex contour integral representations of the derivatives , which can be derived from the complex contour integral representation given above for the Euler–Frobenius polynomials.
|[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)|
Euler-Frobenius polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler-Frobenius_polynomials&oldid=17228