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W_{(1,1)}(t,v)=\frac{-t^{-2k}v^k}{3}(\frac{v^{\frac{3}{2}}-v^{\frac{-3}{2}}}{t^{\frac{3}{2}}-t^{\frac{-3}{2}}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})+\frac{t^{-2k}v^k}{4}(\frac{v-v^{-1}}{t-t^{-1}})^2+\frac{t^{-2k}v^k}{12}(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})-\frac{t^{-k}v^k}{4}(\frac{v^2-v^{-2}}{t^2-t^{-2}}) + \frac{t^{-k}v^k}{8}(\frac{v-v^{-1}}{t-t^{-1}})^2+\frac{t^{-k}v^k}{4}(\frac{v-v^{-1}}{t-t^{-1}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^2-\frac{t^{-k}v^k}{8}(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^4+\frac{-v^kt^{k}}{4}(\frac{v^2-v^{-2}}{t^2-t^{-2}})+\frac{v^kt^{k}}{3}(\frac{v^{\frac{3}{2}}-v^{\frac{-3}{2}}}{t^{\frac{3}{2}}-t^{\frac{-3}{2}}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})+\frac{v^kt^{k}}{8}(\frac{v-v^{-1}}{t-t^{-1}})^2-\frac{v^kt^{k}}{4}(\frac{v-v^{-1}}{t-t^{-1}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^2+\frac{v^kt^{k}}{24}(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^4
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