de Sitter space and Holography

For any function f(P) one can seel 2 (∇ 2 − m 2 )f(P) = (1 − P 2 ) d2 f df− PddP2dP − m2 l 2 fTherefore one has[(1 − P 2 ) d2dP 2 − Pd ddP − m2 l 2 ]G(P(X,Y )) = 0Which as solution in terms of hypergeometric functionsG = const.F(h + , h − , d 2 , 1 + P2)whereh ± = 1 [(d − 1) ± √ ](d − 1)22 − 4m 2 l 2Since the above equation is symmetric under P → −P thereis another solutionG = const.F(h + , h − , d 2 , 1 − P2)One parameter family of dS invariant Green functioncorresponding to a linear combination of these solutions.G α (X,Y ) = 〈α|φ(X)φ(Y )|α〉27