Zeros of Taylor polynomials of (1+z)^p

This is post is related to Extremal Taylor polynomials where it was important to observe that the Taylor polynomials of the function {(1+z)^{-1/2}} do not have zeros in the unit disk. Let’s see how far this generalizes.

The function {f(z)=(1+z)^{-1}} has the rare property that all zeros of its Taylor polynomial have unit modulus. This is clear from

{\displaystyle T_n(z) = \sum_{k=0}^n (-z)^k = (1-(-z)^{n+1})/(1+z)}.

p-33
The Taylor zeros of (1+z)^(-1)

In this and subsequent illustrations, the zeros of the first 50 Taylor polynomials are shown as blue dots, with the unit circle in red for reference.

When the exponent is less than -1, the zeros move inside the unit disk and begin forming nice patterns in there.

p-43
(1+z)^(-4/3)
p-63
(1+z)^(-2)
p-93
(1+z)^(-3)
p-123
(1+z)^(-4)

When the exponent is strictly between -1 and 1, the zeros are all outside of the unit disk. Some of them get quite large, forcing a change of scale in the image.

p-23
(1+z)^(-2/3)
p-13
(1+z)^(-1/3)
p13
(1+z)^(1/3)
p23
(1+z)^(2/3)

Why does this happen when the exponent approaches 1? The function {1+z} is its own Taylor polynomial, and has the only zero at -1.  So, when {p\approx 1}, the Taylor polynomials are small perturbations of {1+z}. These perturbations of coefficients have to create additional zeros, but being small, they require a large value of {z} to help them.

For a specific example, the quadratic Taylor polynomial of {(1+z)^p} is {1 + pz + p(p-1)z^2/2}, with roots {(1\pm \sqrt{(2-p)/p})/(1-p) }. When {p\approx 1}, one of these roots is near {-1} (as it has to be) and the other is large.

Finally, when {p>1} and is not an integer, we get zeros on both sides of the unit circle. The majority of them are still outside. A prominent example of an interior zero is {-1/p} produced by the first-degree polynomial {1 + pz}.

p43
(1+z)^(4/3)
p73
(1+z)^(7/3)
p103
(1+z)^(10/3)

Another related post: Real zeros of sine Taylor polynomials.

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