Today I would like to share another amazing property (convergence) of Digamma Functions. I found this one while playing with some integrals and their series expansion :
For all \(x > 0\)
Great work by my friends, Ronak and Pratik.
So, here' my method.
We'll compute the following sum in two different ways
We can convert the above riemann sum into integral as :
Also, we can compute the above sum by noticing that
Because the integral is independent of the sum, we can interchange the sum and integral as
Separate the above sum as
Now, remember that by the definition of digamma function, we have
Thus, we can conclude that
My other proof was exactly same as Ronak's, which makes use of recurrence function of digamma function.
Furthermore, I make a conjecture here that the above property holds true for some complex too but I'm still working on it's proof. Any help would be appreciated.