Explicit expression for the generating function counting Gesselʼs walks

Gesselʼs walks are the planar walks that move within the positive quadrant by unit steps in any of the following directions: West, North-East, East and South-West. In this paper, we find an explicit expression for the trivariate generating function counting the Gesselʼs walks with k⩾0 steps, which start at (0,0) and end at a given point .