Moments of the Generating Operator of Two Copies of L(F₂)'s over L(F₁)

064<p type="texpara" tag="Body Text" et="abstract" bin="clone" >In this paper, we will consider an example of a (scalar-valued) moment series, under the compatibility. Suppose that we have an amalgamated free product of free group algebras, $L(F_{2})*_{L(F_{1})}L(F_{2})=L(<a,b>)*_{L(<h>)}L(<c,d>)$ 064<p type="texpara" tag="Body Text" et="abstract" bin="clone" >We will provide the method how to find the moment series of $a+b+a^{-1}+b^{-1}+c+d+c^{-1}+d^{-1}.$ Amalgamated freeness of $a+b+a^{-1}+b^{-1}$ and $c+d+c^{-1}+d^{-1}$ over $L(F_{1})$ is used and some combinatorial functions (to explain the recurrence relations) are used to figure out the $n$-th moment of this element.