Hi Sir,

I watched your video on changing the order of a double sum and I think I saw a correlation between it and another function.
Are you familiar with the Stirling numbers of the first kind?
__https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind__
I found an explicit formula for them which is quite messy:
Which seems tangentially related to one of the ones posted on Wikipedia. I’d be happy to share the derivation of this formula with you if you were so inclined.
This formula has a lot of sums, and is virtually unusable unless it is possible to simplify it.
Do you know of any way to simplify k different sums where what is being summed has a function of all k variables? I’m not sure how to approach it.
I do not know if there is anything to be easily found there so I kept looking. I noticed a correlation between the gamma function and it’s derivatives and the different columns of the Stirling numbers.
The first column of the Stirling numbers is equal to n!, the gamma function. The second column is equal to n! times the digamma function + the Euler marcheroni constant. I have no clue, however, to find the third column and subsequent columns because I do not know any sums that are equal to the further polygamma functions. I do not even know if they will differ from the Stirling numbers by some factor of the Euler mascheroni constant or any other numbers.
Is there any way to find a series expression of the polygamma functions? I have no clue how to go about it.
I also know that it is possible to express the polygamma functions in terms of the reimann function which has many analytically continuous forms, it may then be possible to express the Stirling numbers analytically continuated for all complex numbers - that is a big if though!
I’d love to hear your thoughts on this, surely there is some sort of general correlation to be found between the Stirling numbers and the polygamma functions?
I’m not sure how to approach it but I was hoping you would.
Keep making awesome math videos!
Cheers
Jet