Enumeration of the Motzkin paths above a line of rational slope

We introduce L m -Motzkin paths similar to m -Dyck paths. For fixed positive integer m , L m - Motzkin paths are a generalization of Motzkin paths that start at ( 0 , 0 ) , use steps U = ( 1 , 1 ) , D = ( 1 , - 1 ) and L = ( 1 , 0 ) , remain weakly above the line y = m - 1 m x , and end on this line. The number M ( m ) n of L m -Motzkin paths running from ( 0 , 0 ) to ( mn , ( m - 1 ) n ) is called the n -th L m -Motzkin number. We first prove that the generating function M m ( t ) of the L m - Motzkin numbers satisfies the equation M m ( t ) = 1 + tM m ( t ) m + t 2 M m ( t ) 2 m . We then use this generating function and Riordan array to discuss the enumerations of the partial L m - Motzkin paths and the partial grand L m -Motzkin paths. Finally, we consider the colored L m - Motzkin paths, and present two classes of gamma -positive polynomials via enumerating these paths. (c) 2024 Elsevier B.V. All rights reserved.