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\begin{eqnarray*}
A(\lambda,\mu,\eta,t) & = &  \sum_{n=0}^{\mbox{T}} \, \sum_{m = -n}^{n}  X(n,m) (\eta,t) P_{n}^{m}(\mu) e^{im\lambda}
\end{eqnarray*}

\begin{eqnarray*}
A(\lambda,\mu,\eta,t) & = &  \sum_{m=-\mbox{T}}^{\mbox{T}} \, \sum_{n = |m|}^{\mbox{T}}  X(n,m) (\eta,t) P_{n}^{m}(\mu) e^{im\lambda}
\end{eqnarray*}