We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights w ( x ) ≔ w ( x , t ) = e − t / x x α ( 1 − x ) β , t ≥ 0 , defined for x ∈ [ 0 , 1 ] . If t = 0 , this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For t > 0 , the factor e − t / x induces an infinitely strong zero at x = 0 . With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are particular Painlevé V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant, D n ( t ) ≔ det ( ∫ 0 1 x i + j e − t / x x α ( 1 − x ) β d x ) i , j = 0 n − 1 , satisfies the Jimbo–Miwa–Okamoto σ -form of the Painlevé V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new.