= Polynomial construction of hydrogen S states
{title2=$c_{j+1}=[2b(j+1)-a]c_j/[(j+1)(j+2)]$}
For a spherically symmetric bound state, write $\psi=e^{-br}\sum c_jr^j$. The radial Coulomb equation gives the displayed recurrence. Choosing $b=a/(2N)$ terminates it after degree $N-1$, producing a nonzero continuous polynomial times an exponentially decaying function. Its three-dimensional norm $4\pi\int_0^\infty r^2|\psi|^2dr$ is finite and can be normalized. The energy is $-\hbar^2a^2/(8mN^2)$. Continuity at the origin is compatible with the Coulomb cusp; no differentiability as a Cartesian radial function at the origin is asserted.
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