A binomial-coefficient identity arising from the middle discrete series of SU(2,2)

The aim of this paper is to give an elementary proof of certain identities on binomials and state an answer to Remark 8.2 in Takahiro Hayata, Harutaka Koseki, and Takayuki Oda, Matrix coefficients of the middle discrete series of SU(2,2), J. Funct. Anal. 185 (2001), 297-341.