As this question stands about the relationship between 'FunctionAdvisor' and 'convert', and it seems that 'convert' is not able to make the possible conversions listed by FunctionAdvisor(specialize, ...), the following command 'fas' (FunctionAdvisor Specialize) seems a posible workaround:


fas:=(f,g)->subs(op(1,FunctionAdvisor(specialize, f,g)),f):
fas(HeunB(alpha,0,0,0,z),BesselI);
BesselI(1/4*alpha,1/2*z^2)/((1/2*z^2)^(1/4*alpha))*GAMMA(1+1/4*alpha)*2^(1/4*alpha)*exp(1/2*z^2);
fas(HeunB(alpha,0,0,0,2*z),WhittakerM);
WhittakerM(0,1/4*alpha,4*z^2)*exp(2*z^2)/((4*z^2)^(1/2+1/4*alpha)); So, it works in some simple cases. It needs more testing.

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