The Hankel Determinants from a Singularly Perturbed Jacobi Weight

We study the Hankel determinant generated by a singularly perturbed Jacobi weight w ( x , s ) : = ( 1 − x ) α ( 1 + x ) β e − s 1 − x , x ∈ [ − 1 , 1 ] , α > 0 , β > 0 s ≥ 0 . If s = 0 , it is reduced to the classical Jacobi weight. For s > 0 , the factor e − s 1 − x induces an infinitely strong zero at x = 1 . For the finite n case, we obtain four auxiliary quantities R n ( s ) , r n ( s ) , R ˜ n ( s ) , and r ˜ n ( 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 n ( s ) . By variable substitution and some complicated calculations, we show that the quantity R n ( 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 2 s is finite, the scaled Hankel determinant can be expressed by a particular P I I I ′ .