= Hecke prime-power recurrence from q-shift operators
{c}
{title2=$T_pT_{p^r}=T_{p^{r+1}}+p^{k-1}T_{p^{r-1}}$}
On a <Fourier series> put $U_p(\sum a_nq^n)=\sum a_{pn}q^n$ and $V_pf(q)=f(q^p)$. These auxiliary operators need not individually preserve the level-one <modular form> space. Nevertheless $U_pV_p=1$, and the <Fourier coefficients of a composite-index Hecke operator> give $T_{p^r}=\sum_{j=0}^rp^{j(k-1)}V_p^jU_p^{r-j}$. Multiplication by $T_p=U_p+p^{k-1}V_p$, using $U_pV_p=1$, gives the displayed recurrence. In particular the distinction between $U_pV_p=1$ and $V_pU_p\ne1$ is essential.
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