A HIGH-SCHOOL ALGEBRA^{1} , WALLET-SIZED PROOF, OF THE BIEBERBACH CONJECTURE [After L. Weinstein]

Shalosh B. Ekhad^{2} and Doron Zeilberger^{2}

Dedicated to Leonard Carlitz^{3} , master of formal mathematics.

Weinstein’s[2] brilliant short proof of de Branges’[1] theorem can be made yet much shorter(modulo routine calculations), completely elementary (modulo Löwner theory), self contained(no need for the esoteric Legendre polynomials’ addition theorem), and motivated(ditto), as follows. Replace the text between p. 62, line 7 and p. 63, line 7, by Fact 1 below, and the text between the last line of p.63 and p.64, line 7, by Fact 2 below. FACT 1: Let f_{t}(z) = e^{t}z exp(∑_{k=1}^{∞} c_{k}(t)z^{k}) where c_{k}(t) are formal functions of t. Let z and w be related by z/(1 − z)^{2} = e^{t}w/(1 − w)^{2}. The following formal identity holds. (For any formal Laurent series f(z), CT_{z}f(z) denotes the Constant Term of f(z).)

(1 + w)\frac{d}{dt}\{∑_{k=1}^{∞}(4/k − kc_{k}(t)\overline{c_{k}(t)})w^{k}\} =

(1−w) ∑_{k=1}^{∞} Re CT_{z} \{ \frac{\frac{∂f_{t}(z)}{∂t}}{\frac{z∂f_{t}(z)}{∂z}} · (2(1 + ∑_{r=1}^{k} rc_{r}(t)z^{r}) − kc_{k}(t)z^{k}) · (2(1 + ∑_{r=1}^{k} r\overline{c_{r}(t})z^{−r}) − k\overline{c_{k}(t)}z^{−k})\} w^{k}

Proof: Routine. (Even for a human.)

FACT 2: The coefficients A_{k,n}(c) in the formal power series (Laurent in w) expansion (1 − z(2c + (1 − c)(w + 1/w)) + z^{2})^{−1} = ∑_{n=0}^{∞} ∑_{k=0}^{n} A_{k,n}(c)(w^{k} + w^{−k})z^{n} are non-negative for 0 ≤ c ≤ 1.

Proof: This follows immediately from the stronger fact that the coefficients B_{k,n}(c), defined by the expansion (1 − z(2c + (1 − c)(w + 1/w)) + z^{2})^{−1/2} = ∑_{n=0}^{∞} ∑_{k=0}^{n} B_{k,n}(c)(w^{k} + w^{−k})z^{n} can be expressed as L_{k,n}(c)^{2}, for some double sequence L_{k,n}, 0 ≤ k ≤ n, that is real for c in [0, 1]. First use Maple to output B_{k,n}(c), for 0 ≤ k ≤ n ≤ 20, factor them, and observe that for this range they are expressible as L_{k,n}(c)^{2}. Using the gfun Maple package^{4} , Maple conjectures a certain second-order linear recurrence, in n, satisfied by the L_{k,n}(c), and using gfun once again, it computes the linear recurrence satisfied by the squares of the terms of the solution-sequence of the previous recurrence. That new recurrence turns out to be identical with the third-order recurrence, in n, that the WZ method^{5} outputs for B_{k,n}. It follows that indeed B_{k,n} = L_{k,n}^{2}, where now L_{k,n}(c) denotes the solution of the above-mentioned second-order recurrence, with the obvious initial values at n = k, k + 1.

1 and high-school (purely formal) calculus.

2 Department of Mathematics,  Temple University,  Philadelphia,  PA 19122,  USA.

ekhad@math.temple.edu , zeilberg@math.temple.edu . Supported in part by the NSF. We would like to thank Richard Askey, Ira Gessel, Jeff Lagarias and Herb Wilf for comments, that improved readability.

3 L. Carlitz was, for many years, editor of the Duke Journal, until he was relieved from his duties by the proponents

of so-called “modern math”, who despised formal, Carlitz-style, mathematics. This paper makes explicit the power of formal math, that is (too) subtly hidden in Weinstein’s original presentation.

4 Developed by Salvy and Zimmerman, Available by anonymous ftp from ftp.inria.fr in directory lang/maple/INRIA/gfun

.

5 Wilf and Zeilberger, Invent. Math. 108(1992), 575-633. The program, and the input file for this problem, are

available via anonymous ftp to math.temple.edu, in directory pub/zeilberger/programs.

References

1. L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154(1985), 137-152.

2. L. Weinstein, The Bieberbach Conjecture, Inter. Math. Res. Notices (of the Duke J.) 3(1991, No. 5), 61-64.

Revised Version: Nov. 1993. Original Version: Nov. 1992.