Bring Radical - Series Representation

Series Representation

A Taylor series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation can be rewritten as ; by setting, the desired solution is .

The series for can then be obtained by reversion of the Taylor series for (which is simply ), giving:

where the absolute values of the coefficients are sequence A002294 in the OEIS. The series confirms that is odd. This gives

The series converges for |z| < 1 and can be analytically continued in the complex plane. The above result can be written in hypergeometric form as:

Compare with the hypergeometric functions that arise in Glasser's derivation and the method of differential resolvents below.