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\begin{document}

\title[{Some identities for Chebyshev polynomials}]
{Some identities for Chebyshev polynomials}
\author[P. J. Grabner]{Peter J. Grabner$^\dag$}
\address[P. Grabner]{Institut f\"ur Mathematik A,
Technische Universit\"at Graz,
Steyrergasse 30,
8010 Graz,
Austria}
\email{grabner@weyl.math.tu-graz.ac.at}
\thanks{\dag{}This author is supported by the 
 START-project Y96-MAT of the
  Austrian Science Foundation.\\
\hglue\parindent\ddag{}This author's research was done during a visit
  in Graz.}

\author[H. Prodinger]{Helmut Prodinger$^\ddag$}
\address[H. Prodinger]{
The John Knopfmacher
Centre for Applicable Analysis and Number Theory\\
 Department of Mathematics\\
University of the Witwatersrand, P.~O. Wits\\ 
2050 Johannesburg, South Africa}
\email{helmut@gauss.cam.wits.ac.za}

\keywords{Chebyshev polynomials, identities}
\subjclass{Primary: 33C45; Secondary: 05A15, 11B37}
\date{\today}
\begin{abstract}
We prove a generalization of a 
conjectured formula of Melham
and provide some background about the involved (Chebyshev) polynomials.
\end{abstract}

\maketitle

\section{Introduction}
In \cite{Melham99} Melham considered the two sequences
\begin{align*}
U_n&=pU_{n-1}-U_{n-2}, &&U_0=0,\ U_1=1,\\
V_n&=pV_{n-1}-V_{n-2}, &&V_0=2,\ V_1=p,\\
\end{align*}
and conjectured the formula
\begin{align*}
U_n^{2k}+U_{n+1}^{2k}=\sum_{r=0}^k\frac{D^rV_k}{r!}\,U_{n}^{k-r}U_{n+1}^{k-r},
\end{align*}
where $D$ means differentiation with respect to $p$. We remark here that up to
simple changes of variable these polynomials are Chebyshev polynomials. More
precisely 
\begin{align*}
U_n(p)&=\mathcal{U}_{n-1}(\tfrac p2),\\
V_n(p)&=2\mathcal{T}_n(\tfrac p2),
\end{align*}
where $\mathcal{T}_n$ and $\mathcal{U}_n$ denote the classical Chebyshev
polynomials of first and second kind, respectively.

The aim of this paper is to prove a general identity that contains
Melham's conjecture as a special case: Set $W_n=aU_n+bV_n$ and
$\Omega=a^2+4b^2-b^2p^2$, then


\begin{align}\label{weltformel}
W_n^{2k}+W_{n+1}^{2k}=\sum_{r=0}^k \Omega^{k-r}\,\la_{k,r}\,W_{n}^{r}W_{n+1}^{r}, 
\end{align}
with
$$\la_{k,r}=\sum_{0\le 2j\le r}(-1)^j\frac{k(k-1-j)!}{(k-r)!j!
(r-2j)!}p^{r-2j}
$$
and $\la_{0,0}=2$.

From \cite{Andrews_Askey_Roy99,Erdelyi_Magnus_Oberhettinger+53} we know
explicit expansions for Chebyshev polynomials:
\begin{align*} V_j=\sum_{0\le 2k\le j}(-1)^k\binom{j-k}{k}
\frac{j}{j-k}\;p^{j-2k}
\end{align*}
for $j\ge1$ and $V_0=2$.  Then we have
$$
\la_{k,r}=\frac{D^{k-r}V_k}{(k-r)!},
$$
which links Melham's conjecture and \eqref{weltformel}.


\section{Proof of the Formula}




In order to prove \eqref{weltformel} we form 
the generating function
$$g(z)=\sum_{k=0}^\infty z^k\sum_{r=0}^k\Omega^{k-r}\,\la_{k,r}\,\sigma^{r}$$
with $\sigma=W_nW_{n+1}$. We reorder this to obtain
(setting $r-k=t$)
\begin{align*}
g(z)&=\sum_{j\geq1}\sum_{k\geq2j}\sum_{t\geq0}z^t\Omega^t(\sigma z)^k
(-1)^j\frac{(k+t)(k+t-1-j)!}{t!j!(k-2j)!}p^{k-2j}\\
&+\sum_{k\geq0}\sum_{t\geq0}
z^t\Omega^t(\sigma z)^k\frac{(k+t)!}{t!k!}p^k+1\\
&=\sum_{j\geq1}\sum_{k\geq2j}(-1)^jp^{k-2j}(\sigma z)^k
(1-\Omega z)^{j-k-1}
\frac{(k-j-1)!}{j!(k-2j)!}(k-j\Omega z)\\
&+\sum_{k\geq0}(p\sigma z)^k(1-\Omega z)^{-k-1}+1\\
&=\sum_{j\geq1}(-1)^j(\sigma z)^{2j}(1-\Omega z)^{-j-1}
\big(2-(\Omega+p\sigma)z\big)\\
&+\frac1{1-(\Omega+p\sigma)z}+1\\
&=-\frac{(\sigma 
z)^2\big(2-(\Omega+p\sigma)z\big)}{\big(1-(\Omega+p\sigma)z\big)
\big(1-(\Omega +p\sigma)z+\sigma^2z^2\big)}+\frac1{1-(\Omega+p\sigma)z}+1\\
&=\frac{2-(\Omega+p\sigma)z}{1-(\Omega+p\sigma)z+\sigma^2z^2}.
\end{align*}

The generating function of the left hand side is
$$\frac1{1-W_n^2z}+\frac1{1-W_{n+1}^2z}=
\frac{2-(W_n^2+W_{n+1}^2)z}{1-(W_n^2+W_{n+1}^2)z+W_n^2W_{n+1}^2z^2}$$
and the assertion follows from
$$W_n^2+W_{n+1}^2=pW_nW_{n+1}+\Omega,$$
which is easily proved e.~g. by using the explicit
forms 

\begin{align*}
U_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, \qquad
V_n=\alpha^n+\beta^n, 
\end{align*}
with
\begin{align*}
\alpha=\frac{p+\sqrt{p^2-4}}{2},\qquad
\beta=\frac{p-\sqrt{p^2-4}}{2}.
\end{align*}








\section{Further identities}


Many other similar formul\ae{} seem to exist; we just give one
other example;
set
\begin{align*}
a_{k,r}&=\sum_{0\le \la\le r}(-1)^\la p^{2k-2\la}
\frac{k(k-\lfloor\frac \la2\rfloor-1)!2^{\lceil\frac 
\la2\rceil}}{(k-r)!\la!(r-\la)!}
\prod_{i=0}^{\lfloor\frac \la2\rfloor-1}\Big(2k-2\Big\lceil\frac 
\la2\Big\rceil-1-2i\Big)
\end{align*}
and $a_{0,0}=2$, then

%\begin{align*}
%a_{r,k}&=\sum_{0\le 2j\le r}p^{2r-4j}
%\frac{r(r-j-1)!2^j}{(r-k)!(2j)!(k-2j)!}\prod_{i=0}^{j-1}(2r-2j-1-2i)\\
%&-\sum_{0\le 2j+1\le r}p^{2r-4j-2}
%\frac{r(r-j-1)!2^{j+1}}{(r-k)!(2j+1)!(k-1-2j)!}\prod_{i=0}^{j-1}(2r-2j-3-2i)
%\end{align*}



\begin{align*}
W_n^{2k}+W_{n+2}^{2k}=\sum_{r=0}^k\Omega^{k-r}\,a_{k,r}\,W_{n}^{r}W_{n+2}^{r}.
\end{align*}
The proof is as before. 


\bibliographystyle{amsplain}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{1}

\bibitem{Andrews_Askey_Roy99}
G.~Andrews, R.~Askey, and R.~Roy, \emph{Special functions}, Encyclopedia of
  Mathematics and its Applications, no.~71, Cambridge University Press, 1999.

\bibitem{Erdelyi_Magnus_Oberhettinger+53}
A.~Erd{\'e}lyi, W.~Magnus, F.~Oberhettinger, and F.~G. Tricomi, \emph{Higher
  transcendental functions}, vol.~{I}, McGraw-Hill Book Company, Inc., New
  York-Toronto-London, 1953.

\bibitem{Melham99}
R.~S. Melham, \emph{On sums of powers of terms in a linear recurrence},
  Portugali{\ae} Math. \textbf{56} (1999), 501--508.

\end{thebibliography}

%\bibliography{mehlham}
\end{document}


