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.