# Integral relation with $U \left[a,1,z \right]$

1. Jan 6, 2016

### Domdamo

Dear Community,

I get the following relation with the help of Wolfram Mathematica:

$$U\left[a,1,z\right] = \frac{1}{\Gamma\left[a\right]^2\Gamma\left[1-a\right]} \int_{0}^{1} U\left[1,1,zk\right]k^{a-1}(1-k)^{-a}dk$$

I would like to justify this identity in order to use in my article. I do not find such integral representation for the $U\left[a,b,z\right]$ confluent hypergeometric function of the second kind where the integration limits are from $0$ to $1$. I searched for idea in these literature:
Slater, L.J. (1960). Confluent hypergeometric functions. Cambridge University Press.
Bateman, H. Erdelyi, A. (1953). Higher Transcendental Functions. Vol 1. McGraw-Hill.
Abramowitz, M., Stegun, I. (1970). Handbook of Mathematical Functions. Dover.

The only relation which I found, which would be useful is the equation
$$U\left[1,1,z\right]=e^{z}\Gamma[0,z] .$$

Could someone give me a hint how can I justify this relation or which identity is worth to try?

I would appreciate any ideas or hint.

2. Jan 11, 2016