We study the Hankel determinant generated by a singularly perturbed Jacobi weight (Formula presented). If s = 0, it is reduced to the classical Jacobi weight. For s > 0, the factor (Formula presented) induces an infinitely strong zero at x = 1. For the finite n case, we obtain four auxiliary quantities R (s), r (s), ˜R (s), and ˜r (s) by using the ladder operator approach. We show that the recurrence coefficients are expressed in terms of the four auxiliary quantities with the aid of the compatibility conditions. Furthermore, we derive a shifted Jimbo–Miwa–Okamoto σ-function of a particular Painlevé V for the logarithmic derivative of the Hankel determinant D (s). By variable substitution and some complicated calculations, we show that the quantity R (s) satisfies the four Painlevé equations. For the large n case, we show that, under a double scaling, where n tends to ∞ and s tends to 0, such that τ:= n s is finite, the scaled Hankel determinant can be expressed by a particular P.