The confluent hypergeometric function $$_1F_1(a,b,x)$$ satisfies(adsbygoogle = window.adsbygoogle || []).push({});

$$

e^{-x}{}_1F_1(a,b,x)=_1F_1(b-a,b,-x)

$$

By using this relation (and that the complex conjugate of the hypergeometric function is the same as taking the complex conjugate of all its arguments), it can be easily shown that the function

$$

f(z)=e^{-iz/2}{}_1F_1(b/2+ic,b,iz)

$$

is equal to its complex conjugate. My question is if there is some other representation for this function f? Is it equal to some other "famous" special function? Since it is a purely real function, I would imagine that there might exist some representation of it that does not involve complex numbers.

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# A Complex parameters of 1F1 resulting in real values

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