A study on three distinct time-dependent orthogonal polynomials

In this paper, we discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with respect to the weight functions [Formula: see text], where [Formula: see text]. In each case, [Formula: see text], [Formula: see text] and [Formula: see text] possess specific value ranges. Utilizing the ladder operator approach, we derive certain equations for the recurrence coefficients [Formula: see text] and [Formula: see text] from three compatibility conditions. Specifically, when [Formula: see text] and [Formula: see text], we formulate a difference equation for [Formula: see text] that connects it to the [Formula: see text]. Furthermore, we derive first-order (Toda evolution) and second-order differential equations for the [Formula: see text] and [Formula: see text], establishing the connection with the Painlevé IV equation when [Formula: see text]. Additionally, we further investigate the asymptotic behavior of the recurrence coefficients as [Formula: see text]. Following this, for each of the three cases, we establish bi-confluent Heun equations and demonstrate the transformation of the Heun equation into a Painlevé IV equation when [Formula: see text]. Subsequently, we explore the properties of the zeros of the orthogonal polynomials with the given weight function. Finally, we consider the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by the weight function, specifically when [Formula: see text].