= Solution
A <hot band> starts from a thermally populated vibrationally excited state, for example $v=1\to2$. It can satisfy $\Delta v=1$ and need not have a small transition moment, but its lower-state population is reduced by the <Boltzmann distribution>:
$$
\frac{N_1}{N_0}\simeq\exp[-(E_1-E_0)/(k_BT)].
$$
It is therefore usually weak at low temperature and grows relative to the fundamental on heating. <Anharmonicity> shifts its frequency away from the $0\to1$ transition; in a polyatomic molecule, excitation in another mode can also produce a hot-band sequence.
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