Quintic trinomial root

This post looks at an exercise from Special Functions, exercise 6 in Appendix E.

Suppose that x^m+1 + ax − b = 0. Show that

$x = \frac{b}{a} - \frac{b^{m+1}}{a^{m+2}} + \frac{2m + 2}{2!} \frac{b^{2m+1}}{a^{2m+3}} - \frac{(3m+2)(3m+3)}{3!} \frac{b^{3m+1}}{a^{3m+4}} + \cdots$

Use this formula to find a solution to x⁵ + 4x + 2 = 0 to four decimal places of accuracy. When m = 0 this series reduces to the geometric series. Write this sum as a hypergeometric series.

There are several things I find interesting about this exercise.

It’s a possibly useful result.
It’s an application of Lagrange’s inversion formula; that’s where the series comes from.
The equation has m+1 roots. What’s special about the root given by the series?
The sum is not obviously hypergeometric.
What happens when the sum diverges? Is it a useful asymptotic series? Does the analytic continuation work?

I want to address the last two points in this post. Maybe I’ll go back and address the others in the future. To simplify things a little, I will assume m = 4.

The parameters of the hypergeometric series are not apparent from the expression above, nor is it even apparent how many parameters you need. It turns out you need m upper parameters and m-1 lower parameters. Since we’re interested in m = 4, that means we have a hypergeometric function of the form ₄F₃.

$x = \frac{b}{a}\,\, {}_4 F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2}, \frac{3}{4}, \frac{5}{4} ; -\frac{3125b^4}{256a^5}\right)$

We can evaluate this expression in Mathematica as

    f[a_, b_] := (b/a) HypergeometricPFQ[
        {1/5, 2/5, 3/5, 4/5}, 
        {1/2, 3/4, 5/4}, 
        -3125 b^4 / (256 a^5)
    ]

When we evaluate f[4, -2] we get -0.492739, which is the only real root of

x⁵ + 4x + 2 = 0.

Recall that the sign on b is negative, so we call our function with b = -2.

Now lets, try another example, solving for a root of

x⁵ + 3x − 4 = 0.

If we plug a = 3 and b = 4 into the series at the top of the post, the series diverges immediately, not giving a useful asymptotic series before it diverges.

The series defining our hypergeometric function diverges for |z| > 1, and we’re evaluating it at z = −3125/243. However, the function can be extended beyond the unit disk by analytic continuation, and when we ask Mathematica to numerically evaluate the root with by running

    N{ f[3, 4] ]

we get 1, which is clearly a root of x⁵ + 3x − 4.

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