Hypergeometric Solutions for the $q$-Painleve Equation of type $E^{(1)}_6$ by Pade Method

The $q$-Painlev'e equation of type $E^{(1)}_6$ is obtained by Pad'e method. Special solutions in determinant formula to the $q$-Painlev'e equation is presented. A relation between Pad'e method and B"acklund transformation of type $E^{(1)}_6$ is given.

💡 Research Summary

The paper presents a novel derivation of the $q$‑Painlevé equation of type $E^{(1)}_6$ using the Padé approximation method and constructs explicit hypergeometric solutions in determinant form. The authors begin by recalling that $q$‑Painlevé equations are nonlinear $q$‑difference equations whose integrable structure is often revealed through algebraic geometry, Lax pairs, or Bäcklund transformations. However, these approaches can be technically demanding, prompting the search for a more elementary analytic technique.

In the first part, the Padé method is introduced as a way to approximate a given function by a rational function $R(z)=P(z)/Q(z)$, where $P$ and $Q$ are polynomials of degree $N$. By choosing $P$ and $Q$ such that $R(z)$ satisfies the same $q$‑shift relation as a known basic solution built from $q$‑shifted factorials (the $q$‑sigma and $q$‑gamma functions), the authors obtain a rational expression that fulfills a bilinear $q$‑difference relation. This relation can be rewritten as a fractional linear transformation \