$ Bi(z)=\sqrt{3}[c_1f(z)+c_2g(z)] \qquad f(z)=\sum_{0}^\infty 3^k\left( {\frac{1}{3}}\right)_k\frac{z^{3k}}{(3k)!} \qquad g(z)=\sum_{0}^\infty 3^k\left( {\frac{2}{3}}\right)_k\frac{z^{3k+1}}{(3k+1)!} \qquad $