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Suppose that I have an analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ which converges on some disk around the origin.

For a particular function I encountered, I wished to prove that every coefficient, $a_n$, is non-negative.

I am wondering what complex analytic methods exist to detect negative coefficients if all my coefficients are real. What nice ways are there to check if all of the coefficients of my power series are the same sign?

In more generality, are there methods which detect whether eventually all of the coefficients are of the same sign? (That is, whether or not there exists $N$ such that for all $n,m>N$, $a_n$ and $a_m$ will be the same sign)

I am really interested in any, and many, thoughts on this problem. What strategies could possibly work?


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@user8268: Yes, for this problem they are real. I edited to correct it. Although that raises another question: Is there a way to detect whether or not all of the coefficients of my power series are real? –  Eric Naslund May 7 '11 at 20:42
Would induction work? $f(0)=a_0$. Then differentiate to get $a_1$ and so on. –  Fredrik Meyer May 7 '11 at 20:45
@Eric Naslund: (a trivial remark:) reality of coefficients is equivalent to reality of $f(z)$ for $z$ real, or to $f(\bar{z})=\overline{f(z)}$ - but I'm not sure what you imagine you know about $f$ –  user8268 May 7 '11 at 20:54
You've got to give more detail. It would be surprising if there were tools for this that would work in every situation. For illustration, I have published a paper in which the difference of two series with integer coefficients is shown to have non-negative coefficients. So, the provenance of your series is really the whole story here. –  Will Jagy May 7 '11 at 21:06
Reminds me... if $f(z)$ has "oscillatory" behavior, that precludes the possibility that all the $a_n$ be nonnegative, no? –  J. M. May 8 '11 at 1:50

4 Answers 4

If your coefficients are integers, an easy way is to find a combinatorial interpretation. This problem, and problems related to it, appear to be hard to answer algorithmically in general: it is not even known how to detect algorithmically if a coefficient of a rational function is ever zero. See this blog post by Terence Tao on the Skolem-Mahler-Lech theorem.

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If all coefficients are positive, then the dominant singularity lies on the positive real axis.

This would give a criterion to weed out some non-positive series. Of course, it is useless for a proof of positivity.

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Here are some scattered (and mostly obvious) thoughts about the problem (it's very far from an answer).

  • You can use the formula $a_n = n! \, f^{(n)}(0)$.

  • The space of such functions is stable by derivation, sum (positive linear combinations) product and composition.

  • Examples of such functions include $e^x$, $\frac1{1-x}$ and $\tan(x)$ (each exhibits different features). More generally, if $g : \mathbb{C} \to \mathbb{C}$ takes $\mathbb{R}_+$ to $\mathbb{R}_+$, then solutions of $y' = g(y)$, $y(0) \ge 0$ have positive Taylor expansions.

  • The behaviour on the positive (real) numbers is very simple to study. The function is positive, increasing, convex... There's no positive root (you might probably rule out roots on bigger set with additional conditions) and $\underset{R}{\lim} f = + \infty$.

  • A good start might be to study polynomials with positive coefficients (stability by product might be of some help although I doubt it's sufficient here). Also interesting would be rational fractions.

  • Studying $f$ on a circle would lead to a Fourier Series with positive coefficients so both problems might be linked.

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To add: if for some reason you are unable or unwilling to compute derivatives of $f$ symbolically, using Cauchy's differentiation formula might be an interesting alternative. –  J. M. May 8 '11 at 0:38
@ J. M. : Yes that's an interesting idea. Part of it (namely Cauchy on a circle) is encapsulated in the Fourier coefficients. –  Joel Cohen May 8 '11 at 0:58

See slides 37-40 in this talk by Manuel Kauers here

Also, about some series with integer coefficients, a selection from the dissertation of Ken Ono here

The typical situation for showing that a number is integrally represented by a positive integral quadratic form is to express the theta series as a difference of some natural series, then show that the result is positive for large enough $n,$ as in your question. A seminal result is in Duke and Schulze-Pillot, Representations of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990)

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