Special polynomials associated with rational solutions of the fifth Painlevé equation

In this paper special polynomials associated with rational and algebraic solutions of the fifth Painlevé equation (PV) are studied. These special polynomials defined by second-order, bilinear differential-difference equations which are equivalent to Toda equations. The structure of the zeroes of these special polynomials, which involve a parameter, is investigated and it is shown that these have an intriguing, symmetric and regular structure. For large negative values of the parameter the zeroes have an approximate triangular structure. As the parameter increases the zeroes coalesce for certain values and eventually for large positive values of the parameter the zeroes also have an approximate triangular structure, though with the orientation reversed. In fact, the interaction of the zeroes is “solitonic” in nature since the same pattern reappears, with its orientation reversed.