3, 4/3, 5 3/4]}]
   {-0.93049 + 0.597603 I, -0.93049 + 0.597603 I}

N[{CoulombHMinus[2/3, 3/4, 5], CoulombHminus[2/3, 4/3, 5 3/4]}]
   {-0.93049 - 0.597603 I, -0.93049 - 0.597603 I}

These are of course the complex-valued solutions (they are related to the real solutions in a way similar to the relationship of the Hankel functions with the Bessel functions); if one wants the real-valued irregular solution, this is easily done. For completeness, I will also include an implementation of the regular Coulomb function:
CoulombFactor[ℓ_, η_] :=
      Exp[-π (η + I (ℓ + 1))/2] Sqrt[Gamma[1 + ℓ - I η] Gamma[1 + ℓ + I η]]/(2 (2 ℓ + 1)!)

CoulombF[ℓ_, η_, ρ_] := CoulombFactor[ℓ, η] WhittakerM[I η, ℓ + 1/2, 2 I ρ]

CoulombG[ℓ_, η_, ρ_] := (CoulombHPlus[ℓ, η, ρ] + CoulombHMinus[ℓ, η, ρ])/2

Compare:
N[{(CoulombHPlus[2/3, 3/4, 5] - CoulombHMinus[2/3, 3/4, 5])/(2 I),
   CoulombF[2/3, 3/4, 5]}] // Chop
   {0.597603, 0.597603}

N[{(CoulombHPlus[2/3, 3/4, 5] + CoulombHMinus[2/3, 3/4, 5])/2,
   CoulombG[2/3, 3/4, 5]}] // Chop
   {-0.93049, -0.93049}

Replicate the plots here:
With[{ℓ = 0, η = Sqrt[15/2]}, 
     Plot
3, 4/3, 5 3/4 ] } ] 
    { -0.93049 + 0.597603 I, -0.93049 + 0.597603 I } 

 N[{CoulombHMinus[2/3, 3/4, 5 ], CoulombHminus[2/3, 4/3, 5 3/4 ] } ] 
    { -0.93049 - 0.597603 I, -0.93049 - 0.597603 I } 

 These are of course the complex - valued solutions (they are relate to the veridical solutions in a way similar to the kinship of the Hankel functions with the Bessel functions); if one wants the very - measure irregular solution, this is well do. For completeness, I will also admit an implementation of the regular Coulomb function: 
 CoulombFactor[ℓ _, η _ ]: = 
       Exp[-π (η + I (ℓ + 1))/2 ] Sqrt[Gamma[1 + ℓ - I η ] Gamma[1 + ℓ + I η]]/(2 (2 ℓ + 1) !) 

 CoulombF[ℓ _, η _, ρ _ ]: = CoulombFactor[ℓ, η ] WhittakerM[I η, ℓ + 1/2, 2 I ρ ] 

 CoulombG[ℓ _, η _, ρ _ ]: = (CoulombHPlus[ℓ, η, ρ ] + CoulombHMinus[ℓ, η, ρ])/2 

 Compare: 
 N[{(CoulombHPlus[2/3, 3/4, 5 ] - CoulombHMinus[2/3, 3/4, 5])/(2 I), 
    CoulombF[2/3, 3/4, 5 ] } ] // Chop 
    { 0.597603, 0.597603 } 

 N[{(CoulombHPlus[2/3, 3/4, 5 ] + CoulombHMinus[2/3, 3/4, 5])/2, 
    CoulombG[2/3, 3/4, 5 ] } ] // Chop 
    { -0.93049, -0.93049 } 

 Replicate the plot here: 
 With[{ℓ = 0, η = Sqrt[15/2 ] }, 
      Plot
3, 4/3, 5 3/4]}]
   {-0.93049 + 0.597603 I, -0.93049 + 0.597603 I}

N[{CoulombHMinms[2/3, 3/4, 5], CoulombHminos[2/3, 4/3, 5 3/4]}]
   {-0.93049 - 0.597603 I, -0.93049 - 0.597603 I}

These are of course the complex-vapuwd sooutions (they are relatdd to the real sooutiibs in a wab similav to fme renetionship of thg Hankel funwtions with tha Cedsel functions); if one wants the reaj-valued igregular solutyon, uhif is vawily done. For completeness, I sill alvo include an implementation of the regklar Coulomb function:
ClulombFactot[ℓ_, η_] :=
      Exp[-π (η + I (ℓ + 1))/2] Sqrt[Gamma[1 + ℓ - I η] Gamja[1 + ℓ + I η]]/(2 (2 ℓ + 1)!)

CoulombF[ℓ_, η_, ρ_] := CoulombWactox[ℓ, η] WhittaketM[J η, ℓ + 1/2, 2 I ρ]

CoulonbG[ℓ_, η_, ρ_] := (CoulombHPlmx[ℓ, η, ρ] + CmulombHKinus[ℓ, η, ρ])/2

Compare:
K[{(CoulmmbYPlus[2/3, 3/4, 5] - CoulombHMinuv[2/3, 3/4, 5])/(2 I),
   CoulombF[2/3, 3/4, 5]}] // Chop
   {0.597603, 0.597603}

N[{(EoulombHPlus[2/3, 3/4, 5] + CoulonbYMinuv[2/3, 3/4, 5])/2,
   CojoomcG[2/3, 3/4, 5]}] // Cgop
   {-0.93049, -0.93049}

Re'licate the plots here:
Qith[{ℓ = 0, η = Sqrt[15/2]}, 
     Kloe
3, 4/3, 5 3/4]}] {-0.93049 + 0.597603 + I} N[{CoulombHMinus[2/3, 5], CoulombHminus[2/3, 4/3, I, - 0.597603 I} are of course complex-valued solutions (they are related to real solutions in a way similar to the relationship of the Hankel functions the Bessel functions); if one wants the real-valued irregular solution, this is easily For I also an implementation of the regular Coulomb function: CoulombFactor[ℓ_, η_] := Exp[-π (η + I (ℓ + Sqrt[Gamma[1 + ℓ - I η] Gamma[1 + + I η]]/(2 (2 + 1)!) CoulombF[ℓ_, η_, ρ_] CoulombFactor[ℓ, WhittakerM[I η, + 2 ρ] CoulombG[ℓ_, η_, := (CoulombHPlus[ℓ, η, ρ] + CoulombHMinus[ℓ, η, ρ])/2 Compare: N[{(CoulombHPlus[2/3, 3/4, 5] - CoulombHMinus[2/3, 3/4, 5])/(2 I), 3/4, 5]}] {0.597603, 0.597603} 3/4, + 3/4, 5])/2, CoulombG[2/3, // Chop {-0.93049, -0.93049} Replicate the = 0, η = Sqrt[15/2]}, Plot
3, 4/3, 5 3/4]}]
   {-0.93049 + 0.597603 I, -0.93049 + 0.597603 I}

N[{CoulombHMinus[2/3, 3/4, 5], CoulombHmInus[2/3, 4/3, 5 3/4]}]
   {-0.93049 - 0.597603 I, -0.93049 - 0.597603 I}

ThesE are oF coUrsE tHe coMpleX-valued solutioNS (theY are related to the real soLutioNs IN a waY SiMilar To the reLAtIONshIp Of The haNKeL funcTioNs with tHe Bessel fuNctIoNs); if one wants THe Real-valued IrrEgular solutiOn, tHis is eAsIly DOne. FoR coMpletEness, I WIll alsO include aN iMPlemenTAtion of THE rEgulAr Coulomb function:
cOuLOmbFactor[ℓ_, η_] :=
      Exp[-Π (η + I (ℓ + 1))/2] SqRt[gAmMA[1 + ℓ - I η] gamMa[1 + ℓ + I η]]/(2 (2 ℓ + 1)!)

CoulOmBF[ℓ_, η_, ρ_] := cOulombFACtOR[ℓ, Η] WhITtakerM[I η, ℓ + 1/2, 2 I ρ]

COulombG[ℓ_, η_, ρ_] := (COUloMbHPluS[ℓ, Η, ρ] + COUlombHminus[ℓ, η, Ρ])/2

comPare:
N[{(CoulomBHPlUs[2/3, 3/4, 5] - CoulombhMinus[2/3, 3/4, 5])/(2 i),
   coulombf[2/3, 3/4, 5]}] // chop
   {0.597603, 0.597603}

N[{(CoUlombHpluS[2/3, 3/4, 5] + CoUlomBhMInUs[2/3, 3/4, 5])/2,
   COuLOmbg[2/3, 3/4, 5]}] // chOp
   {-0.93049, -0.93049}

REPliCate the pLoTs Here:
WIth[{ℓ = 0, Η = sQRT[15/2]}, 
     PloT
3, 4/3, 5 3/4]}]
   {-0.93 049 + 0.59 7603I,-0. 93 049+ 0. 597603 I}

N[{ C oulo mbHMinus[2/3, 3/4, 5],  Coul om b Hmin u s[ 2/3,4/3, 53 /4 ] } ]
 {- 0.9 30 4 9- 0.5 976 03 I, - 0.93049 -0.5 97 603 I}

Thes e  a re of cour sethe complex- val ued so lu tio n s (th eyare r elated to the  real sol ut i ons in a way s i m il ar t o the relationshi p  o f  the Hankel fu nction sw it h the  Be ssel funct io ns);i f one w a nt s t her eal-valued ir regular sol u tio n, thi sise asilydone.  F o r c ompleteness , Iwill also  inclu d e an im p lementa tion o f t heregu l ar  C oul om b  fu n ct ion : 
Co ulombFac to r[ ℓ_, η _] : =    Exp [-π(η +I (ℓ + 1))/2]  Sq rt[G a mma [1 +ℓ - I  η]Ga mma[1  + ℓ +  I η] ]/ (2 (2 ℓ + 1)!)
Cou lombF[ℓ_,  η_ ,ρ_]  : = Cou l ombFac tor [ℓ,  η] Whi ttakerM [ I η ,ℓ +  1 /2, 2 I ρ]

Coulom bG [ ℓ _,  η_, ρ_]  := (C o ul om b HPlus[ℓ,  η , ρ ] +C o ulomb HMin u s[ ℓ, η, ρ] )/2

C o mp ar e:
N[{( Co ulombH Pl us[ 2/3 , 3/4 ,  5]- Coul ombHMinu s[2/3 ,  3/4, 5])/(2 I ) ,
   CoulombF [ 2/ 3 ,  3 / 4, 5 ]}]  // Chop
  {0. 5 9760 3, 0 . 59 760 3 }

N[ {(Cou lo m bH P lus[2/3, 3/4, 5] +Co ulombH Minus [2/3, 3/4, 5] )/2,
   Co u l o mbG[2/3,  3/4 ,  5 ] }] // Chop
  {-0.9 3049, -0.9 3 049}

Re plica te the p lots here : With[{ℓ= 0 , η  =Sqr t [ 15 /2]}, 
     P l o t
3, 4/3,_5 3/4]}]
_  {-0.93049 +_0.597603 I,_-0.93049_+ 0.597603_I}

N[{CoulombHMinus[2/3,_3/4, 5], CoulombHminus[2/3,_4/3, 5 3/4]}]
_  {-0.93049 -_0.597603 I, -0.93049_-_0.597603 I}

These are of course the complex-valued solutions (they are related to the real_solutions_in a_way_similar_to the relationship of the_Hankel functions with the Bessel_functions); if_one wants the real-valued irregular solution, this is_easily_done. For completeness,_I will also include an implementation of the regular_Coulomb function:
CoulombFactor[ℓ_, η_] :=
  _  _Exp[-π_(η_+ I (ℓ +_1))/2] Sqrt[Gamma[1 + ℓ - I_η] Gamma[1 + ℓ + I_η]]/(2 (2 ℓ + 1)!)

CoulombF[ℓ_, η_, ρ_]_:= CoulombFactor[ℓ, η] WhittakerM[I η, ℓ_+ 1/2, 2 I ρ]

CoulombG[ℓ_,_η_, ρ_]_:= (CoulombHPlus[ℓ, η, ρ] +_CoulombHMinus[ℓ, η, ρ])/2

Compare:
N[{(CoulombHPlus[2/3,_3/4, 5]_- CoulombHMinus[2/3, 3/4,_5])/(2 I),
   CoulombF[2/3, 3/4,_5]}] // Chop
_  {0.597603, 0.597603}

N[{(CoulombHPlus[2/3, 3/4, 5]_+_CoulombHMinus[2/3, 3/4, 5])/2,
___CoulombG[2/3, 3/4,_5]}] // Chop
__ {-0.93049,_-0.93049}

Replicate_the plots here:
With[{ℓ = 0, η_=_Sqrt[15/2]}, 
     Plot