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%\section{preamble}
\title[Orthogonality of Gegenbauer polynomials]
{The orthogonality of Gegenbauer polynomials $\boldsymbol{C_n^\la(x)}$ 
for $\boldsymbol{\la<-n}$  }

\author{Kathy Driver and Helmut Prodinger}

\address{Kathy Driver,  Helmut Prodinger,
The John Knopfmacher
Centre for Applicable Analysis and 
Number Theory,
 School of Mathematics,
University of the Witwaters\-rand, P.~O. 
Wits, 
2050 Johannesburg, South Africa, email:
{\tt 036kad@cosmos.wits.ac.za},
{\tt helmut@gauss.cam.wits.ac.za}.
}


\date{\today}


\begin{document}


\maketitle

The Gegenbauer (or ultraspherical) polynomials $C_n^\la(z)$
can be defined  by the generating function
\begin{equation*}   
(1-2zr+r^2)^{-\la}=\sum_{n=0}^\infty C_n^\la(z)\,r^n;
\end{equation*} 
see \cite{AnAsRo99}. For $\la>-1/2$, these polynomials
$C_n^\la(z)$ are \textsl{orthogonal\/} on the interval $(-1,1)$ with
respect to the \textsl{weight function\/} $(1-z^2)^{\la-1/2}$ 
and therefore have all of their zeros in that interval. As the 
parameter $\la$ decreases below $-1/2$, the zeros of
 $C_n^\la(z)$ depart from $(-1,1)$ and when $\la <1-n$,
all the zeros of   $C_n^\la(z)$ lie on the imaginary
axis (cf.~\cite{DrDu01}). 

An interesting question is whether there is some underlying
orthogonality that gives rise to the location of the zeros
of  $C_n^\la(z)$ on the imaginary axis for $\la<1-n$. We show
that this is indeed the case for  $\la<-n$ by proving that
the sequence of polynomials 
$\big\{i^nC_n^\la(ix)\big\}_{n=1}^\infty$ 
is orthogonal on $(-\infty,\infty)$ with respect to the
weight function $(1+x^2)^{\la-1/2}$.
 Note that for a given parameter $\la$, 
the set of polynomials is finite ($1\le n<-\la$). This
orthogonality property does not appear to hold for $-n< \la <1-n$.
Since $C_n^\la(z)$ is an even/odd function of $z$    according
to whether $n$ is  even/odd, the sequence of polynomials
$\big\{i^nC_n^{\la}(ix)\big\}$ is real--valued  for $x$ real, while
the weight function
$(1+x^2)^{\la-1/2}$ is positive for all $x$. 
It therefore
follows from the classical theory of real orthogonal polynomials
that all zeros of $C_n^\la(ix)$  are real for $\la<-n$. Thus
the zeros  of $C_n^\la(z)$  are pure imaginary
 for $\la<-n$. 

As an application of the orthogonality of  
$\big\{i^nC_n^\la(ix)\big\}$   for $\la<-n$, we derive an 
expression for
\begin{equation*}   
\int_{-\infty}^{\infty}C_l^\la(ix)\,C_m^\la(ix)\,C_n^\la(ix)
\,(1+x^2)^{\la-1/2}\,dx 
\end{equation*}  
which is analogous to the result given in 
\cite[p.~321, eq. (6.8.10)]{AnAsRo99} for $\la>-1/2$. 

\begin{theorem}   \label{th1}
For $n,m<-\la $  we have 


\begin{equation} \label{eq1}  
\int_{-\infty}^{\infty}i^n\,C_n^\la(ix)\,i^m\,C_m^\la(ix)\,
\,(1+x^2)^{\la-1/2}\,dx =g_n\delta_{nm}
\end{equation}  
with
\begin{equation}   
g_n=\frac{(-1)^n2^{-2\la-1}\big[(\la)_n\big]^2
\big[\Gamma(-n-\la)\big]^2}{n!\,(n+2\la)_n\,\Gamma(-2n-2\la)}.
\end{equation}
\end{theorem}

\begin{proof}
The \textsl{Rodrigues formula\/} for the Gegenbauer polynomials
is given by (cf.~\cite[p.~303, eq. (6.4.14)]{AnAsRo99})


\begin{equation}   \label{eq3}
(1-x^2)^{\la-1/2}C_n^{\la}(x)=\frac{(-2)^n(\la)_n}{n!\,(n+2\la)_n}
\,D^n\left[(1-x^2)^{n+\la-1/2}\right].
\end{equation}
%
Replacing $x$ by $ix$ yields
\begin{equation}   \label{eq4}
(1+x^2)^{\la-1/2}C_n^{\la}(ix)=\frac{(-2)^n(\la)_n}{n!\,(n+2\la)_n\,
i^n}
\,D^n\left[(1+x^2)^{n+\la-1/2}\right].
\end{equation}
%
 Then 

\begin{multline} \label{eq5}
\int_{-\infty}^{\infty}i^n\,C_n^\la(ix)\,i^m\,C_m^\la(ix)\,
\,(1+x^2)^{\la-1/2}\,dx\\
= \frac{(-2)^n(\la)_n}{n!\,(n+2\la)_n}
\int_{-\infty}^{\infty}\,\Big\{D^n\big[(1+x^2)^{n+\la-1/2}\big]
\Big\}
\,i^m\,C_m^{\la}(ix)\,dx.
\end{multline}
%
Integrating the right hand side of (\ref{eq5}) by parts $n$ 
times, each time differentiating $i^mC_m^{\la}(ix)$ and
integrating the expression in curly brackets, we observe
that at the $k^{\textrm{th}}$  stage, the integrated part
$\pm\{D^{n-k}(1+x^2)^{n+\la-1/2}\}
D^{k-1}(i^mC_m^{\la}(ix))$ has limit value zero as $x$ tends
to $\pm\infty$, since $\la<-n$. After $n$ such integrations by 
parts we therefore have

\begin{multline} \label{eq6}
\int_{-\infty}^{\infty}i^n\,C_n^\la(ix)\,i^m\,C_m^\la(ix)\,
\,(1+x^2)^{\la-1/2}\,dx\\
= \frac{(-1)^n(-2)^n(\la)_n}{n!\,(n+2\la)_n}
\int_{-\infty}^{\infty}(1+x^2)^{n+\la-1/2}
\,D^n\big(i^m\,C_m^{\la}(ix)\big)\,dx.
\end{multline}
%
If $m\neq n$, we may choose $n$ to be larger (or interchange
$m$ and       $n$ in the preceding argument) and conclude that
\begin{equation*}   
\int_{-\infty}^{\infty}i^n\,C_n^\la(ix)\,i^m\,C_m^\la(ix)\,
\,(1+x^2)^{\la-1/2}\,dx=0,
\end{equation*} 
because $C_m^{\la}(ix)$ is a polynomial of degree $m$. 

For $m=n$, the right hand side of (\ref{eq6}) is equal to 
\begin{equation*}   
\frac{2^n(\la)_n}{n!\,(n+2\la)_n}
\int_{-\infty}^{\infty}(1+x^2)^{n+\la-1/2}
\,D^n\big(i^n\,C_n^{\la}(ix)\big)\,dx=g_n.
\end{equation*} 
%
Now (cf.~\cite[p.~277]{Rainville60}) 
\begin{equation*}   
D^n\big[C_n^\la(x)\big]=2^n(\la)_n,
\end{equation*}
%
so it follows that 
\begin{equation} \label{eq7} 
g_n=\frac{(-1)^n2^{2n}\big[(\la)_n\big]^2}{n!\,(n+2\la)_n}
\int_{-\infty}^{\infty}(1+x^2)^{n+\la-1/2}\,dx.
\end{equation}
%
To evaluate the integral, we use the \textsl{beta integral
on a halfline}, (\cite[p.~7, eq. (1.1.20)]{AnAsRo99})
\begin{equation*}   
\int_0^\infty\frac{s^{x-1}}{(1+s)^{x+y}}\,ds=
\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}:
\end{equation*}
\begin{align*}
\int_{-\infty}^{\infty}(1+x^2)^{n+\la-1/2}\,dx&=
2\int_{0}^{\infty}(1+x^2)^{n+\la-1/2}\,dx\\
&=\int_{0}^{\infty}(1+t)^{n+\la-1/2}\,t^{-1/2}\,dt
=\frac{\sqrt{\pi}\,\Gamma(-n-\la)}{\Gamma(-n-\la+\frac12)}.
\end{align*}
%
From this the formula for $g_n$ is immediate using the duplication
formula for the Gamma function.
\end{proof}


\textbf{Corollary.}   For $l,m,n<-\la$ 
\begin{multline*}   
\int_{-\infty}^{\infty}C_l^\la(ix)\,C_m^\la(ix)\,C_n^\la(ix)
\,(1+x^2)^{\la-1/2}\,dx \\
=\frac{(-1)^s2^{-2\la-1}\,(\la)_{s-l}\,(\la)_{s-n}\,
(\la)_{s-m}\,(2\la)_{s}\,(\la)_l\,(\la)_{l+1}
\big[\Gamma(-l-\la)\big]^2}
{(s+\la)\,(s-l)!\,(s-n)!\,(s-m)!\,(\la)_{s}\,(2\la)_{2l}
\,\Gamma(-2l-2\la) }
\end{multline*}
where $l+m+n=2s$ is even and the sum of any two of $l$, $m$, $n$
is not less than the third. The integral is zero is all other cases.    


\begin{proof}   
The \textsl{linearization formula\/}
 for the product of two Gegenbauer polynomials is given by
(cf.~\cite[p.~319, eq. (6.8.4)]{AnAsRo99})
\begin{equation}\label{eq9}   
C_m^{\la}(x)\,C_n^{\la}(x)=\sum_{k=0}^{\min(m,n)}a(k,m,n)
\,C_{m+n-2k}^{\la}(x)
\end{equation}
%
where
\begin{equation*}   
a(k,m,n)=\frac{(m+n+\la-2k)\,(\la)_k\,(\la)_{m-k}\,
(\la)_{n-k}\,(2\la)_{m+n-k}\,(m+n-2k)!}
{(m+n+\la-k)\,k!\,(m-k)!\,(n-k)!\,(\la)_{m+n-k}\,(2\la)_{m+n-2k} }.
\end{equation*}
%
From (\ref{eq9}) it follows  that 
\begin{multline}\label{eq11}   
\int_{-\infty}^{\infty}i^l\,C_l^\la(ix)\,i^m\,
C_m^\la(ix)\,i^n\,C_n^\la(ix)
\,(1+x^2)^{\la-1/2}\,dx \\
=\sum_{k=0}^{\min(m,n)}a(k,m,n)
\int_{-\infty}^{\infty}i^{m+n}\,
C_{m+n-2k}^\la(ix)\,i^l\,C_l^\la(ix)\,(1+x^2)^{\la-1/2}\,dx 
\end{multline}
%
where by Theorem~\ref{th1} the integral on the right hand side of
(\ref{eq11}) is zero for $l\neq m+n-2k$. 
Hence only the term with $k=s-l=m+n-s$ survives in the sum,
and the right hand side of (\ref{eq11}) reduces to 
\begin{align*}   
a(m+{}&{}n-s,m,n)\,i^{m+n-l}\int_{-\infty}^{\infty}
\Big[i^l\,C_l^\la(ix)\Big]^2\,(1+x^2)^{\la-1/2}\,dx\\
&=a(m+n-s,m,n)(-1)^{s-l}g_l\\
&=
\frac{(l+\la)\,(\la)_{s-l}\,(\la)_{s-n}\,
(\la)_{s-m}\,(2\la)_{s}\,l!}
{(s+\la)\,(s-l)!\,(s-n)!\,(s-m)!\,(\la)_{s}\,(2\la)_{l} }
\frac{(-1)^s2^{-2\la-1}\big[(\la)_l\big]^2
\big[\Gamma(-l-\la)\big]^2}{l!\,(l+2\la)_l\,\Gamma(-2l-2\la)},
\end{align*}
where we have used (\ref{eq1}). This is equivalent to the formula
given in the corollary.
\end{proof}

\bibliographystyle{amsplain}
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\begin{thebibliography}{1}
\bibitem{AnAsRo99}
G.~Andrews, R.~Askey, and R.~Roy, \emph{Special functions}, Encyclopedia of
  Mathematics and its Applications, vol.~71, Cambridge University Press, 1999.

\bibitem{DrDu01}
K.~Driver and P.~Duren, \emph{Zeros of ultraspherical polynomials and the
  {Hilbert--Klein} formulas}, J. Comput. Appl. Math. \textbf{135} (2001),
  293--301.

\bibitem{Rainville60}
E.~Rainville, \emph{Special functions}, Chelsea, New York, 1960.

\end{thebibliography}


\end{document}

