Orthogonal Polynomials With a Semi-Classical Weight and Their Recurrence Coefficients

Focusing on the weight function
$\omega (x,t)=x^{\alpha }e^{-\frac {1}{3}x^{3}+tx}, x\in [0,\infty),\,\,\,\,\alpha >-1,\,\,\,\,t> 0$
, we state its asymptotic orthogonal polynomials. Through Toda evolution, differential equations of
$\alpha _{n}(t)$
and
$\beta _{n}(t)$
have been worked. Consequently, we also talk about the approximate value of
$\alpha _{n}(t)$
. Basing on the asymptotic value of
$\alpha _{n}(t)$
, the asymptotic of second order differential equation of
$P_{n}(z)$
and expansion of the logarithmic form of Hankel determinant are confirmed.