More "fun" with the polygamma function.
EDIT: Show that .
Then use the result to find .
I'm well aware of that integral. But does ?
And I thought . Nevermind. (Giggle)
For , evaluate
Not too difficult, but the result is somewhat surprising if you've never seen it.
This thread has started as "Prove some identities", and has become "Use this transcendental function's properties as proven in Whittaker and Watson". There's nothing wrong with it, but it'd be fun to go back to some relatively simple stuff.
Here's an easy problem I made up : show that, for any positive integer , we have
I didn't mean to kill this thread! Somebody must have solved the problem I gave but forgotten to post their solution... right? ^^