LATEX

$$Bi(z)=\sqrt{3}\bigl[c_1f(z) + c_2g(z)\bigr] \\ f(z)=1+\frac{1}{3!}z^3+\frac{1\cdot4}{6!}z^6+\frac{1\cdot4\cdot7}{9!} z^9+\ldots \quad=\sum_{0}^{\infty}3^k (\frac{1}{3})_k \frac{z^{3k}}{(3k)!}\\ g(z)=z+\frac{2}{4!}z^4+\frac{2\cdot5}{7!}z^7+\frac{2\cdot5\cdot8}{10!} z^{10}+\ldots \qquad=\sum_{0}^{\infty}3^k (\frac{2}{3})_k \frac{z^{3k+1}}{(3k+1)!}\\ \bigl(\alpha+\frac{1}{3}\bigr)_0=1\\ 3^k\bigl(\alpha+\frac{1}{3}\bigr)_k=(3\alpha+1)(3\alpha+4)\ldots(3\alpha+3k-2)\\ (\alpha \text{ arbitrary} ; k=1, 2, 3 , \ldots)$$