Refinements of two identities on $(n,m)$-Dyck paths

Rosena R. X. Du, Kuo Yu

For integers $n,m$ with $n\ge 1$ and $0\le m\le n$, an $(n,m)$-Dyck path is a lattice path in the integer lattice $\mathbb{Z}\times \mathbb{Z}$ using up steps $(0,1)$ and d…