$p$-adic heights of generalized Heegner cycles

We relate the $p$-adic heights of generalized Heegner cycles to the derivative of a $p$-adic $L$-function attached to a pair $(f,\chi )$, where $f$ is an ordinary weight $2r$ newform and $\chi$ is an unramified imaginary quadratic Hecke character of infinity type $(\mathrm{\ell },0)$, with $0<\mathrm{\ell }<2r$. This generalizes the $p$-adic Gross-Zagier formula in the case $\mathrm{\ell }=0$ due to Perrin-Riou (in weight two) and Nekov'a\u{r} (in higher weight).