- #1
Domdamo
- 12
- 0
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.
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.