The　relativistic　full　Euler　system　with　generalized　Chaplygin　proper　energy　density–pressure　relation　is　studied.　The　Riemann　problem　is　solved　constructively.　The　delta　shock　wave　arises　in　the　Riemann　solutions,　provided　that　the　initial　data　satisfy　some　certain　conditions,　although　the　system　is　strictly　hyperbolic　and　the　first　and　third　characteristic　fields　are　genuinely　nonlinear,　while　the　second　one　is　linearly　degenerate.　There　are　five　kinds　of　Riemann　solutions,　in　which　four　only　consist　of　a　shock　wave　and　a　centered　rarefaction　wave　or　two　shock　waves　or　two　centered　rarefaction　waves,　and　a　contact　discontinuity　between　the　constant　states　(precisely　speaking,　the　solutions　consist　in　general　of　three　waves),　and　the　other　involves　delta　shocks　on　which　both　the　rest　mass　density　and　the　proper　energy　density　simultaneously　contain　the　Dirac　delta　function.　It　is　quite　different　from　the　previous　ones　on　which　only　one　state　variable　contains　the　Dirac　delta　function.　The　formation　mechanism,　generalized　Rankine–Hugoniot　relation　and　entropy　condition　are　clarified　for　this　type　of　delta　shock　wave.　Under　the　generalized　Rankine–Hugoniot　relation　and　entropy　condition,　we　establish　the　existence　and　uniqueness　of　solutions　involving　delta　shocks　for　the　Riemann　problem.　©　2018,　Springer　International　Publishing　AG,　part　of　Springer　Nature.