▁the ▁C ox eter ▁diagram ▁D 4 , ▁ ▁analog ous ▁to ▁the ▁ 2 4 ▁symmet ries ▁of ▁the ▁hyper ge ometric ▁differential ▁equations ▁obtained ▁by ▁K um mer . ▁The ▁symmet ries ▁fixing ▁the ▁local ▁He un ▁function ▁form ▁a ▁group ▁of ▁order ▁ 2 4 ▁is omorphic ▁to ▁the ▁symmetric ▁group ▁on ▁ 4 ▁points , ▁so ▁there ▁are ▁ 1 9 2 / 2 4 ▁=   8 ▁=   2   ×   4 ▁essentially ▁different ▁solutions ▁given ▁by ▁acting ▁on ▁the ▁local ▁He un ▁function ▁by ▁these ▁symmet ries , ▁which ▁give ▁solutions ▁for ▁each ▁of ▁the ▁ 2 ▁ex ponents ▁for ▁each ▁of ▁the ▁ 4 ▁singular ▁points . ▁ ▁The ▁complete ▁list ▁of ▁ 1 9 2 ▁symmet ries ▁was ▁given ▁by ▁ ▁using ▁machine ▁calculation . ▁Several ▁previous ▁attempts ▁by ▁various ▁authors ▁to ▁list ▁these ▁by ▁hand ▁contained ▁many ▁errors ▁and ▁om issions ; ▁for ▁example , ▁most ▁of ▁the ▁ 4 8 ▁local ▁solutions ▁listed ▁by ▁He un ▁contain ▁serious ▁errors . ▁ ▁See ▁also ▁He ine – St ielt jes ▁polynomials , ▁a ▁general ization ▁of ▁He un ▁polynomials . ▁ ▁References ▁▁ ▁A . ▁Erd ély i , ▁F . ▁Ober het ting er , ▁W . ▁Magn us ▁and ▁F . ▁Tr ic omi ▁High er ▁Trans c endent al ▁functions ▁vol . ▁ 3 ▁( Mc G raw ▁Hill , ▁NY , ▁ 1 9 5 3 ). ▁▁▁▁ ▁H ahn ▁W .( 1 9 7 1 ) ▁On