Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !}$. That is, $$ \begin{aligned} { }_1 F_1(a ; c ; z) & =\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !}, c \neq 0,-1,-2, \ldots \\ & =\frac{\Gamma(c)}{\Gamma(a)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)}{\Gamma(c+n)} \frac{z^n}{n !} . \end{aligned} $$

Other symbols used instead of ${ }_1 F_1(a ; c ; z)$ are $\Phi(a, c ; z)$ and $M(a, c, z)$.

Another function closely related to ${ }_1 F_1(a ; c ; z)$ is the Tricomi function défined by
$$U(a, c, z)=\frac{\Gamma(1-c)}{\Gamma(a-c+1)}{ }_1 F_1(a, c, x)+\frac{\Gamma(c-1)}{\Gamma(a)} x^{1-c} {}_1F_1(a-c+1,2-c, x) .
$$
**My question:** How to prove this formula for $0<x\leq y$
$$\sum^\infty_{k=0}\frac{(xy)^k}{k!\Gamma(1+\alpha+k)}\Gamma(\beta+k)U(\beta+k,1+\alpha+2k,x+y)=\frac{\Gamma(\beta)}{\Gamma(1+\alpha)}{ }_1F_1(\beta,\alpha+1, x) U(\beta,\alpha+1, y)$$

Thank you in advance.

`Gamma[1]/Gamma[2] Hypergeometric1F1[1, 2, 1] HypergeometricU[1, 2, 2]`

and`Gamma[1]/Gamma[2] Hypergeometric1F1[1, 2, 2] HypergeometricU[1, 2, 1]`

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