where the absolute values of the coefficients form sequence A002294 in the OEIS. The radius of convergence of the series is In hypergeometric form, the Bring radical can be written as It may be interesting to compare with the hypergeometric functions that arise below in Glasser's derivation and the method of differential resolvents. Solution of the general quintic The roots of the polynomial can be expressed in terms of the Bring radical as and its four conjugates. The problem is now reduced to the Bring–Jerrard form in terms of solvable polynomial equations, and using transformations involving polynomial expressions in the roots only up to the fourth degree, which means inverting the transformation may be done by finding the roots of a polynomial solvable in radicals. This procedure gives extraneous solutions, but when the correct ones have been found by numerical means, the roots of the quintic can be written in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions (defined broadly to include Bring radicals) of a single variable — an algebraic solution of the general quintic. Other characterizations Many other characterizations of the Bring radical have been developed, the first of which is in terms of "elliptic transcendents" (related to elliptic and modular functions) by Charles Hermite in 1858, and further methods later developed by other mathematicians. The Hermite–Kronecker–Brioschi characterization In 1858, Charles Hermite published the first known solution to the general quintic equation in terms of "elliptic transcendents", and at around the same time Francesco Brioschi and Leopold Kronecker came upon equivalent solutions. Hermite arrived at this solution by generalizing the well-known solution to the cubic equation in terms of trigonometric functions and finds the solution to a quintic in Bring–Jerrard form: into which any quintic equation may be reduced by means of Tschirnhaus transformations as has been shown. He observed that elliptic functions had an analogous role to play in the solution of the Bring–Jerrard quintic as the trigonometric functions had for the cubic. For and write them as the complete elliptic integrals of the first kind: where Define the two "elliptic transcendents": They can be equivalently defined by infinite series: If n is a prime number, we can define two values and as follows: and When n is an odd prime, the parameters and are linked by an equation of degree n + 1