# A ${}_2 F_1$ equivalent of the Tricomi $U$ function?

The confluent hypergeometric function ${}_1F_1(a;b;z)$ has a natural partner in the Tricomi function $U(a,b,z)$, which provides a second, linearly independent solution to the confluent hypergeometric equation. This is not strictly necessary as one can usually find a second ${}_1F_1$ function with appropriate parameters as a second solution, but the Tricomi function seems to be a more natural choice.

For the gaussian hypergeometric equation, a similar situation holds in that one can find appropriate parameters for two gauss hypergeometric functions ${}_2 F_1(a,b;c;z)$ that will provide a general solution. However, there seems to be no natural partner for ${}_2F_1$ analogous to $U$. Is this the case, or am I missing something? If this is indeed the case because of some fundamental reason, why is it?

In case this is all too general, more precisely: is there a (canonical?) solution $f(a,b;c;z)$ to the hypergeometric differential equation that limits as $$\lim_{b\rightarrow\infty} f(a,b;c;\frac z b)=U(a,c,z)?$$

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