We are concerned with the monic orthogonal polynomials with respect to the modified Gaussian weight \(\begin{aligned} w(x)=w(x;s):=\textrm{e}^{-N[x^2+s(x^6-x^2)]},\qquad x\in \mathbb {R} \end{aligned}\) with parameters \(N> 0\) and \(s\in [0,1]\) . Using the ladder operator approach and associated compatibility conditions, we show that the recurrence coefficient \(\beta _n(s)\) satisfies a nonlinear fourth-order difference equation, which is the second member of the discrete Painlevé I hierarchy. We find that the orthogonal polynomials satisfy a second-order ordinary differential equation, with all the coefficients expressed in terms of \(\beta _n(s)\) . By considering the s evolution, we derive the differential-difference equation for the recurrence coefficient \(\beta _n(s)\) . We also obtain some relations between the Hankel determinant \(D_n(s)\) , the sub-leading coefficient \(\textrm{p}(n,s)\) of the monic orthogonal polynomials and the recurrence coefficient \(\beta _n(s)\) . Finally, we study the large n asymptotics of the recurrence coefficient \(\beta _n(s)\) , the sub-leading coefficient \(\textrm{p}(n,s)\) and the logarithmic derivative of \(D_n(s)\) for fixed \(N>0\) . We also consider the asymptotics of \(\beta _n(s)\) when n/N is fixed as \(n\rightarrow \infty \) .
