Inequalities and asymptotics for a terminating \({}_4F_3\) series
For integers \(n\), \(r\), and \(s\) satisfying \(n>0\) and \(r>s\), define the terminating hypergeometric series \[ \begin{aligned} R(n,s,r) &={}_4F_3\biggl(\begin{gathered} -n, \, n+1, \, -s, \, s+1 \\ 1, \, 1-r, \, 1+r \end{gathered} \biggm| 1 \biggr) \\ &=\sum_{k=0}^n\binom nk\binom{n+k}k \frac{(-s)_k(s+1)_k}{(1-r)_k(1+r)_k}(-1)^k, \end{aligned} \] where \((a)_k=\Gamma(a+k)/\Gamma(a)\). It is conjectured in [\textit{A.~Kresch} and \textit{H.~Tamvakis}, Duke Math. J. 110, No.~2, 359--376 (2001; Zbl 1072.14514)] that \(| R(n,s,r)| \leq1\). Note that the quantity \(R(n,s,r)\) is related to the Racah polynomials \(R_n(x(x+1);0,0,-N-1,N+1)=R(n,x,N+1)\) and the conjecture states that the absolute value of a Racah polynomial is bounded by its value at \(x=0\). Using the properties of the Racah polynomials, which are discrete orthogonal polynomials, the authors give upper bounds for \(R(n,s,r)\); the bounds confirm in several cases the required inequality. The major remaining part of this paper is devoted to the computation of the asymptotics of \(R(n,x,N+1)\); in particular, it (Theorem~3.8) is shown that, for fixed rational \(\lambda\in(0,1]\) and \(\gamma\in(1,+\infty)\), one has \(R(n,\lambda n,\gamma n+1)\to0\) as \(n\to\infty\).
scientific article; zbMATH DE number 5316796