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In this paper we show that the position and the derivative operators, $\hatq$ and $\hat D$, can be treated as ladder operators connecting the variousvectors of two biorthonormal families, $\mathcal{F}_\varphi$ and$\mathcal{F}_\psi$. In particular, the vectors in $\mathcal{F}_\varphi$ areessentially monomials in $x$, $x^k$, while those in $\mathcal{F}_\psi$ are weakderivatives of the Dirac delta distribution, $\delta^{(m)}(x)$, times somenormalization factor. We also show how bi-coherent states can be constructedfor these $\hat q$ and $\hat D$, both as convergent series of elements of$\mathcal{F}_\varphi$ and $\mathcal{F}_\psi$, or using two differentdisplacement-like operators acting on the two vacua of the framework. Ourapproach generalizes well known results for ordinary coherent states.

Bi-coherent states as generalized eigenstates of the position and the momentum operators