= Solution
A \b[<vibrational fundamental>] is the transition $v=0\to1$. A \b[first <vibrational overtone>] is $v=0\to2$. A \b[<hot band>] starts from a thermally populated vibrationally excited level, such as $v=1\to2$.
For a strictly harmonic potential and a dipole linear in displacement, the allowed transitions have $\Delta v=\pm1$. A fundamental and a $1\to2$ hot band then share the same frequency, while the overtone has zero intensity. <Anharmonicity> separates the adjacent gaps and can give overtone intensity; nonlinear displacement dependence of the dipole can also contribute. A hot band is defined by its lower state, not merely by being weak.
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