New formula for 9- j symbols and their direct calculation

An algebraic expression for 9-j symbols in the form of a summation of the products of binomial coefficients is obtained. An algorithm is also devised to calculate these binomial coefficients recursively. This avoids the evaluation of factorials of integers, which is the main source of overflow in calculation of coupling coefficients for large angular momenta. Thus, the new formula permits accurate calculation of 9-j symbols. In addition, it has higher symmetry and involves only a twofold summation. Therefore, a direct approach for accurate and efficient calculation of 9-j symbols for very large angular momenta is thereby established.