The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation (Q2890784)

For the monic polynomials \(p_n(x)\) orthogonal on the half-line with respect to the weight \(w(x)=x^{\alpha}e^{-x^2+tx}\) depending on the constant \(\alpha>-1\) and a real parameter \(t\), the authors prove that the coefficient \(b_n\) in the three-term recurrence relation \(xp_n=a_np_{n+1}+b_np_n+a_np_{n-1}\), as a function of \(t\), satisfies the classical fourth Painlevé equation PIV with the parameters determined by \(\alpha\) and \(n\). The authors present several proofs, one of which is based on the use of the differential-difference system for the recurrence coefficients \(a_n,b_n\), another one uses the ladder operator approach, and the last one utilizes the Lax pair for the orthogonal polynomials. Finally, the authors explain the relation between the mentioned above differential-difference system for the recurrence coefficients and the Bäcklund transformations of PIV.