A $p$-adic Gross-Zagier formula for twisted triple product $p$-adic $L$-functions attached to finite slope families

Our main objective in the present paper is to generalise the work of Blanco-Chacón and Fornea on the $p$-adic Gross-Zagier formula for twisted triple product $p$-aidc $L$-function. We extend their main result to the case of finite slope families of Hilbert modular forms and also allow the prime $p$ to be inert in the real quadratic field $L$.