A Non-Autonomous Discrete Painleve Equation and Orthogonal Polynomials for an Exponential Weight
Over the last few decades, the autonomous discrete Painleve 1 equation has been studied from several different perspectives. One can view the dP1 equation as a Quispel-Roberts-Thompson mapping, where each point in an orbit is gotten from the previous by intersecting and reflecting. Another perspective is that of Okamoto and Sakai, where the dynamics are understood through the geometry of a special surface called the space of initial values. Using these perspectives, one can gain insight into several related combinatorial problems, such as counting 4-valent planar maps, and blossom trees.
In this talk we will discuss an integrable deautonomization of the discrete Painleve 1 equation. This specific equation arises from a recurrence relation for orthogonal polynomials with respect to an exponential weight, and has connections to counting 4-valent maps of any genus. From the Q.R.T. and Okamoto/Sakai perspectives we can understand the non-autonomous dynamics and address questions from the 4-valent map counting problem.