Moments on quadratic binomial products (Q2628022)

From the author's abstract: ``We prove a general transformation theorem that expresses the moments on quadratic products of binomial coefficients as linear sums of their four initial values. Sixteen summation formulae are presented explicitly as examples. They contain, as special cases, the previous results due to \textit{X. Chen} and the author [J. Math. Anal. Appl. 349, No. 2, 311--316 (2009; Zbl 1168.11004)] and \textit{P. J. Miana} and \textit{N. Romero} [Biblioteca Rev. Mat. Iberoam., 203--208 (2008; Zbl 1192.05013); J. Number Theory 130, No. 8, 1876--1887 (2010; Zbl 1214.11033)].'' The paper deals with the expressions \[ \Theta^\delta_\gamma(m,n):=\sum_{k\geq \delta}(k-\delta/2)^\gamma\binom{2m+\delta}{k+k}\binom{2n+\delta}{n+k}, \] where \(\delta\in\{0,1\}\) ans \(\gamma,m,n\in\mathbb{N}\). Then among others, using elementary methods and some preparatory results, it is shown that \[ \Theta_{2\gamma}^0(m,n)=\sum_{j=0}^\gamma \binom{2m+2n-2j}{m+n-j} \sum_{i=0}^j\binom{2m}{i}\binom{2j-2m}{j-i}\frac{(m-i)^{2\gamma+1}}{2m-2j}. \]