LHS[m_, a_, t1_] :=
   NIntegrate[
     HermiteH[2 m + 1, z] E^(-z^2 + Sqrt[2] a z E^(-I t1)), {z, 0, Infinity}]

 RHS[m_, a_, t1_] :=
   -(((-1)^m*(2*m + 1)!!*E^((1/2)*(-I)*t1)*
     E^(((1/2)*a^2)/E^(2*I*t1))*(a^2/E^(2*I*t1))^(m + 1/2)*
     Gamma[-m - 1/2, 0, ((1/2)*a^2)/E^(2*I*t1)])/(2*Sqrt[2]))