introduce the notion of weighted free Motzkin paths which is a lattice path consisting of Motzkin steps without the restrictions that it has to end with a point on the $x$-axis and it does not go below the $x$-axis. We then give a bijection between weighted free Motzkin paths and weighted partial Motzkin paths with an elevation line, which leads to a matrix identity involving the number of weighted partial Motzkin paths and the sequence $(1, k,
k^2, \ldots)$. The idea of the elevation operation is also used by Cameron and Nkwanta in their combinatorial proof of the identity (\[eq2.2\]) in a more restricted form. By extending our argument to weighted partial Motzkin paths with multiple elevation lines, we obtain a combinatorial proof of an identity recently derived by Cameron and Nkwanta, in answer to their question.

We also give a generalization of the matrix identity (\[eq2.2\]) and give a combinatorial proof by using colored Dyck paths.

Riordan Arrays
==============

In this section, we give a brief introduction to the notion of Riordan arrays [@shapiroba; @shapirotr; @sprugnoli]. Let us use (\[eq2.2\]) and (\[eq1.1\]) as examples. Start with two generating functions $g(x)=1+g_1x+g_2x^2+\cdots$ and $f(x)=f_1x+f_2x^2+\cdots$ with $f_1\neq 0$. Let $H=(h_{i,j})_{n,j\geq 0}$ be 