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\centerline{\bf A Footnote On Two Proofs of the Bieberbach-de Branges Theorem}
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\centerline{Herbert S. Wilf\footnote{$^*$}{Supported in part by the United States Office of Naval Research}}
\centerline{University of Pennsylvania}
\centerline{Philadelphia, PA 19104-6395}
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Louis de Branges' original proof [2] of the conjecture of Bieberbach reduced the
problem ultimately to that of showing the nonnegativity of a certain sequence of
polynomials, and then he observed that their nonnegativity had previously
been established by Askey and Gasper [1].
Very recently [6] a short, elegant proof of the theorem
has been given by Lenard Weinstein. This time, via the L\" owner
theory, Weinstein reduced the problem to showing that a certain
sequence of functions is nonnegative, and then he proved the
nonnegativity of these functions, which established the Milin
conjecture, and {\it a fortiori}, the theorem of Bieberbach-de Branges
also.
The purpose of this ``footnote'' is to observe that the functions that
Weinstein encountered, but did not identify, are none other than the
same polynomials of Askey and Gasper that de Branges had met.
In particular, then, Weinstein's argument gives an independent proof
of the nonnegativity of the Askey-Gasper polynomials.
Indeed, Weinstein defines functions $A_{k,n}(c)$ by the generating
function $$f(z)={1\over {1-2z(c+(1-c)\cos{\theta})+z^2}}=\sum_{n\ge
0}\sum_{k=0}^nA_{k,n}(c)z^n\cos{k\theta},\eqno(1)$$ in which we have written
$c$ for his $\cos^2{\phi}$ and $A_{k,n}(c)$ for his $\Lambda_k^n(t)$,
where $\sin{\phi}=e^{-t/2}$.
We compare this with the generating function for the Tschebycheff polynomials $U_n(x)$ of the second kind ([5], p. 301) $${1\over
{1-2xt+t^2}}=\sum_{n=0}^{\infty}U_n(x)t^n,$$ and we find that
$$U_n(c+(1-c)\cos{\theta})=\sum_{k=0}^nA_{k,n}(c)\cos{k\theta}.\eqno(2)$$
We propose to find an explicit formula for the $A_{k,n}(c)$ by finding all of their derivatives, with respect to $c$, evaluated at $c=1$. To do that, we differentiate (2) $r$ times with respect to $c$ and put $c=1$, obtaining
$$(1-\cos{\theta})^rU_n^{(r)}(1)=\sum_{m=0}^nA_{m,n}^{(r)}(1)\cos{m\theta}.$$
Hence if we multiply by $\cos{k\theta }$ and integrate from $0$ to
$2\pi$ we obtain $$\pi
A_{k,n}^{(r)}(1)=2^{r+1}U_n^{(r)}(1)\int_0^{\pi}(\sin{\theta})^{2r}\cos{2k\theta }\, d\theta.$$
Next, for integers $r\ge 0$ and $k$, the integral evaluation
$$\int_0^{\pi}(\sin{t})^{2r}\cos{2kt}\, dt={{(-1)^k\pi}\over {4^r}}{{2r}\choose {r+k}}$$
is easy to check, by induction on $r$. Hence we have
$$\pi A_{k,n}^{(r)}(1)={{\pi (-1)^k (2r)!U_n^{(r)}(1)}\over {2^{r-1}(r-k)!(r+k)!}},$$
and so
$$A_{k,n}^{(r)}(1)={{(-1)^k}\over {2^{r-1}}}{{2r}\choose {r+k}}U_n^{(r)}(1).\eqno(3)$$
To evaluate the derivatives of $U_n$ at 1, we have from [5] p. 277,
that $$U_n(x)={{(n+1)!}\over {({3\over 2})_n}}P_n^{({1\over 2},{1\over
2})}(x)=C_n^{1}(x),\eqno(4)$$ the latter being the Gegenbauer
polynomial. But it has the expansion ({\it ibid}, eq. 15, p. 279)
$$C_n^{1}(x)=\sum_k{{(2)_{n+k}}\over {2^kk!(n-k)!({3\over
2})_k}}(x-1)^k$$ from which we can read off all of its derivatives at
$x=1$, and therefore by (4), we see all of the derivatives of $U_n(x)$
at $x=1$; by (3) we find all of the derivatives of the $A_{n,k}(c)$ at
$c=1$; and by Taylor's theorem we obtain an explicit formula for them,
viz. $$\eqalign{A_{k,n}(c)&=2(-1)^k\sum_r{{2r}\choose {r+k}}{{n+r+1}\choose
{2r+1}}(c-1)^r\cr
&=2(1-c)^k{{n+k+1}\choose {2k+1}}\ _3F_2\left[\matrix{k+{1\over 2},&k+n+2,&k-n;\cr 2k+1,&k+{3\over 2};&&\cr} 1-c\ \right].\cr}$$
These are exactly the polynomials of Askey and Gasper.
If we put together the above with Weinstein's proof, we obtain another representation of $A_{k,n}(c)$ as a (rather unpleasant) sum of squares, viz.
$$\eqalign{A_{k,n}(\cos^2{\phi})&=2\sum_{n'+n''=n}{{(n''-k)!}\over {(n''+k)!}}(P_{n'}(\cos{\phi}))^2(P_{n''}^k(\cos{\phi}))^2
\cr
&\quad +2\bigl\{\sum_{\scriptstyle{{{k'+k''=k}\atop {n'+n''=n}}}}+\sum_{\scriptstyle{{{k'-k''=k}\atop
{n'+n''=n}}}}\bigr\}{{(n'-k')!(n''-k'')!}\over {(n'+k')!(n''+k'')!}}(P_{n'}^{k'}(\cos{\phi}))^2(P_{n''}^{k''}(\cos{\phi}))^2
,\cr}\eqno(5)$$
where the $P_n(x)$ are the Legendre polynomials and the $P_n^k(x)$ are the associated Legendre functions.
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\noindent{\bf Remarks.} I am indebted to Walter Hayman for calling my
attention to the question of identifying the functions that appear in
(1) above, and to the referee for several helpful suggestions. The
identity (2) is a special case of a result of Feldheim [3]. For other
representations of the $A_{k,n}(c)$ as sums of squares, including some
that are related to (5), see Gasper [4].
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\centerline{\bf References}
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{\parindent 12pt
\item{1.} R. Askey and G. Gasper, Positive Jacobi sums, II, {\it Amer. J. Math.} {\bf 98} (1976), 709-737.
\item{2.} L. de Branges, A proof of the Bieberbach conjecture, {\it Acta Math.} {\bf 154} (1985), 137-152.
\item{3.} E. Feldheim, Contributions a la theorie des polynomes de Jacobi, Mat.-Fiz.-Lapok {\bf 48} (1941), 453-504.
\item{4.} G. Gasper, Positivity and special functions, in {\it Theory and Application of Special Functions}, (R. Askey, ed.), Academic Press, New York, 1975.
\item{5.} Earl D. Rainville, Special Functions, MacMillan, New York, 1960.
\item{6.} Lenard Weinstein, The Bieberbach conjecture, International Mathematics Research Notices, 1991 No. 5, pp. 61-64, in {\it Duke Math. J.} {\bf 64} (1991).
}
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