Rational Hadamard products via Quantum Diagonal Operators

We use the remark that, through Bargmann-Fock representation, diagonal operators of the Heisenberg-Weyl algebra are scalars for the Hadamard product to give some properties (like the stability of periodic fonctions) of the Hadamard product by a rational fraction. In particular, we provide through this way explicit formulas for the multiplication table of the Hadamard product in the algebra of rational functions in $\C[[z]]$.

💡 Research Summary

The paper presents a novel operator‑theoretic framework for the Hadamard (coefficient‑wise) product of formal power series, exploiting the Bargmann‑Fock representation of the Heisenberg‑Weyl algebra. In this representation the creation operator $a^\dagger$ acts as multiplication by $z$ and the annihilation operator $a$ as differentiation $\partial_z$. Consequently the number operator $N=a^\dagger a$ is diagonal on the monomial basis ${z^n}{n\ge0}$ with eigenvalue $n$. The authors observe that any function $f(N)$ of this diagonal operator acts on a series $F(z)=\sum{n\ge0}f_n z^n$ simply by scaling each coefficient: $f(N)z^n = f(n)z^n$. This observation translates the Hadamard product into an operator action: for two series $A(z)=\sum a_n z^n$ and $B(z)=\sum b_n z^n$ one has
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