Power solution expansions of the analogue tothe first Painleve equation
  The fourth-order analog to the first Painlev\'{e} equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found. The exponential additions to the expansion of solution near $z=\infty$ are computed. The obtained results confirm the hypothesis that the fourth-order analog of the first Painlev\'{e} equation determines new transcendental functions. By means of the methods of power geometry the basis of the plane lattice is also calculated. 
Aleksandr D. Bruno and Nikolai A. Kudryashov