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\title{\bf Solution to Problem 2 in the SIAM Activity Group on
Orthogonal Polynomials and
Special Functions Newsletter}
\author{by Richard Askey and George Gasper\\
(askey\@@math.wisc.edu \ and \ george\@@math.nwu.edu)}
\date{September 5, 1997 version}
\maketitle

% Solution to Problem 2 by Richard Askey and George Gasper

\centerline{\bf Problem 2}

 Is it true that
\[x^2 t^x {}_2F_1(x+1,x+1;2;1-t) \]
is a convex function of $x$ whenever  $ -\infty <x< \infty$
and $0<t<1?$

\centerline{Submitted by George Gasper, August 19, 1992.}
\vskip.5cm
\centerline{\bf Solution to Problem 2}
\centerline{by Richard Askey and George Gasper}
\centerline{(askey\@@math.wisc.edu \ and \ george\@@math.nwu.edu)}
\vskip.2cm
Let $0<t<1, -\infty <x< \infty,$ and set
\begin{equation}
f_t(x)=x^2 t^x {}_2F_1(x+1,x+1;2;1-t).    \label{eq1}
\end{equation}
\noindent
By using the binomial theorem to expand $t^x=(1-(1-t))^x$ in powers
of $1-t,$ it follows that
\[
f_t(x)=x^2\sum_{j=0}^\infty \sum_{k=0}^\infty
\frac{(-x)_j(1+x)_k(1+x)_k}{j!\, k!\, (2)_k}\, (1-t)^{j+k}\]
\[
=x^2 \sum_{n=0}^\infty 
\frac{(-x)_n}{n!}\,(1-t)^n \,{}_3F_2(-n,1+x,1+x;2,1+x-n;1)
\]

\noindent
after setting $j=n-k$ and changing the order of summation.  Then, 
application of the $b=-n$ case of the transformation formula 
\cite[3.8(1)]{wnb} to the above ${}_3F_2$ series yields the expansion
formula (generating function)
\begin{equation}
f_t(x)=x^2 \sum_{n=0}^\infty F_n(x)\, (1-t)^n     \label{eq2}
\end{equation} 
with
\begin{equation}
F_n(x)={}_3F_2(-n,1+x,1-x;2,1;1), \quad n=0,1,\ldots \, .    \label{eq3}
\end{equation}

 From (\ref{eq3}) it is clear that each $F_n(x)$ is a polynomial
of degree $n$ in $x^2$, and hence, by (\ref{eq2}), that $f_t(x)$ is an even 
function of $x.$  Also, computations of the coefficients of the polynomials
$F_n(x)$ for many values of $n$ suggest that each
 $F_n(x)$ is an absolutely monotonic
function (one whose power series coefficients are nonnegative) of $x.$
  Then (\ref{eq2}) would imply that $f_t(x)$ is an absolutely monotonic 
function of $x$ and, consequently,
 a convex function of $x$ when $0<t<1.$   Thus it suffices to prove
that each $F_n(x)$ is an absolutely monotonic function of $x.$

 First observe that
\begin{equation}
F_n(x)= \frac{1}{n!\,(n+1)!}\, S_n(-x^2;1,1,0), \label{eq4}
\end{equation}
where
\begin{equation}
\frac{S_n(x^2;a,b,c)}{(a+b)_n(a+c)_n}= 
{}_3F_2(-n,a+ix,a-ix;a+b,a+c;1), \label{eq5}
\end{equation}
n=0,1,\ldots , are the continuous dual Hahn polynomials,
  see  \cite[\S 1.3]{ks}.  Since the  polynomials 
$\{S_n(y;a,b,c)\}_{n\ge 0}$
are orthogonal
on the interval $(0,\infty)$ with respect to a positive weight function
 when  $c\ge 0$ and either $a,b>0$ or $a,b$ are complex
conjugates with positive real parts, the zeros $y_{k,n}(a,b,c), 1\le k \le n,$
of each $S_n(y;a,b,c)$ are positive, and hence
\[
 S_n(x^2;a,b,c)=
C_n(a,b,c) \prod_{k=1}^n \big(y_{k,n}(a,b,c)-x^2 \big)\]
where, by inspection of the right side of (\ref{eq5}), $C_n(a,b,c)>0.$
Therefore, with the above-mentioned restrictions on $a,b,c$ each
\begin{equation}
S_n(-x^2;a,b,c)=
C_n(a,b,c) \prod_{k=1}^n \big(y_{k,n}(a,b,c)+x^2 \big) \label{eq6}
\end{equation}
is an absolutely monotonic function of $x$, and hence, by (\ref{eq4}) and
(\ref{eq2}), $F_n(x)$ and $f_t(x)$ are absolutely monotonic 
functions of $x,$  which completes the proof. 
From this proof it is clear that the function $t^x {}_2F_1(x+1,x+1;2;1-t)$
is also an absolutely monotonic, and hence convex, function of $x.$
 For a discussion of the origin of this problem, see page 11 of
the Fall 1994 issue of this Newsletter (note that in the fourth
displayed equation $(K_{i(x+iy)}(a))^2$ should read $(K_{ix}(a))^2 $\,).

\noindent {\bf Remark 1.} Formula (\ref{eq2})  can also be derived by 
first applying the Pfaff-Kummer transformation 
formula  \cite[(1.5.5)]{gr} to the 
${}_2F_1$ series in  (\ref{eq1}) with the additional  restriction
 that $|(1-t)/t|<1,$ using the binomial theorem to expand each of the
$t^{-j-1}$ powers of $t$ in powers of $1-t,$ changing the order of 
summation, and then employing analytic continuation to remove the
 $|(1-t)/t|<1$ restriction.

 
\noindent {\bf Remark 2.}  From (\ref{eq6}) and an extension of the
derivation of (\ref{eq2}) it 
follows that the generating 
function (cf.  \cite[(1.3.6)]{ks}, \cite[p. 398]{lv})
\[(1-t)^{x-c} \,{}_2F_1(a+x,b+x;a+b;t) \]
\[= \sum_{n=0}^\infty\frac{S_n(-x^2;a,b,c)}{ n! \,(a+b)_n}\, t^n, 
\quad 0\le t <1, \]
is an absolutely monotonic 
function of $x$  when  $c\ge 0$ and either $a,b>0$ or $a,b$ are complex
conjugates with positive real parts, and  that the generating 
function (cf. \cite[19.10 (25)]{erd}, \cite[(1.3.9)]{ks})
\[e^t \,{}_2F_2(a+x,a-x;a+b,a+c;-t) \]
\[= \sum_{n=0}^\infty\frac{S_n(-x^2;a,b,c)}{ n! \,(a+b)_n\, (a+c)_n}\, t^n, 
\quad 0\le t <\infty, \]
is an absolutely monotonic function of $x$  when $a,b,c>0.$

\noindent {\bf Remark 3.}  Similarly, additional absolutely monotonic 
functions can be obtained by using the generating functions for
the Wilson $W_n(x^2;a,b,c,d)$ and the Askey-Wilson $p_n(x;a,b,c,d|q)$
polynomials in \cite[\S 1.1 and
\S 3.1]{ks}, and their limit cases,
along with suitable changes in variables.
In particular, it follows from 
\cite[(1.1.2) and (1.1.6)]{ks}, \cite[(1.2) and (2.4)]{wil} that 
the generating function
\[{}_2F_1(a+x,b+x;a+b;t)\, {}_2F_1(c-x,d-x;c+d;t)\]
\[= \sum_{n=0}^\infty\frac{W_n(-x^2;a,b,c,d)}
{ n! \,(a+b)_n\, (c+d)_n}\, t^n, 
\quad 0\le t <1, \]
is an absolutely monotonic 
function of $x$  when  Re$(a,b,c,d)>0$ and non-real parameters occur in
conjugate pairs with $a+b>0$ and $c+d>0.$  

  

\begin{thebibliography}{9}
\bibitem{wnb} Bailey, W.N.: {\em Generalized Hypergeometric Series},
Cambridge University Press, London, 1935: reprinted by
Stechert-Hafner Service Agency, New York--London, 1964.

\bibitem{erd} Erd\'elyi, A. et al.: {\em Higher Transcendental Functions},
Vol. III, McGraw-Hill, New York--London, 1955.

\bibitem{gr} Gasper, G. and Rahman, M.: {\em Basic Hypergeometric Series},
Encyclopedia of Mathematics and Its Applications, Vol. 35, Cambridge 
University Press, 1990.

\bibitem{ks} Koekoek, R. and Swarttouw, R.F.: The Askey-scheme of
hypergeometric orthogonal polynomials and its $q$-analogue, 
Delft University of Technology, Faculty of
    Technical Mathematics and Informatics, Report 94-05, 1994;
    revised version of February 20, 1996 obtainable by anonymous ftp from:
    unvie6.un.or.at, directory {siam/opsf\_new/koekoek\_swarttouw,}
   file {koekoek\_swarttouw1.ps,} or via the World Wide Web:

    ftp://unvie6.un.or.at/siam/opsf\_new/\break 
koekoek\_swarttouw/koekoek\_swarttouw1.ps

\bibitem{lv} Letessier, J. and Valent, G.: Dual birth and death processes
and orthogonal polynomials, SIAM J. Math. Anal. 46 (1986), 393--405.

\bibitem{wil} Wilson, J.A.: Asymptotics for the ${}_4F_3$ polynomials,
J. Approx. Theory. 66 (1991), 58--71.

\end{thebibliography}

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