Originally posted by @humyNot offhand, the ratio of factorials is k-1 permute c and if you were dividing by k! then it would be the binomial expansion coefficient. This might be tractable using some sort of generating function, I'll play around with it, but my intuition is it's not going to simplify.
is there any way to simplify either
( ∑[k=c, p] (k – 1)! / ( k^m (k – c)! ) ) / ∑[k=c, p] ( (k – 1)! / k^(m-1) (k – c)! )
c, m, p ∈ ℕ,
( ∑[k=c, ∞] (k – 1)! / ( k^m (k – c)! ) ) / ∑[k=c, ∞] (k – 1)! / ( k^(m-1) (k – c)! )
c, m ∈ ℕ,
c ∈ [1, m–2],
m ∈ [3, ∞ )
I really hope there is as I use ...[text shortened]... tical except m in the sum for the numerator is replaced with m-1 in the sum for the denominator.
Originally posted by @deepthoughtFortunately, even if it doesn't simplify, I have worked out an efficient work around of evaluating the second one and I have just tested it today and it works albeit at the price of requiring an awful lot of computer memory needed to hold a huge 2D array of new maths constants that I call the "nac constants". It works around ~1000 times faster than using the simple but inefficient 'brute force' method. I will explain this efficient method in my book.
... but my intuition is it's not going to simplify.
Originally posted by @eladarmy field is NOT specifically maths in particular (try AI) and never was and I am NOT a fully qualified mathematician and never will be.
Here you go again asking for some smart person in your field to do your math for you.
Originally posted by @humySo you are trying to do something beyond your abilities and need others to pull it off.
my field is NOT specifically maths in particular (try AI) and never was and I am NOT a fully qualified mathematician and never will be.
I do most of my work myself including any required maths but I don't have the arrogance to think no one is smart enough to help me when I just occasionally get stuck. What is wrong with just occasionally asking smart person ...[text shortened]... who I intend to cite credit in my book thanks to their contributions -what is wrong with that?
Originally posted by @eladarnope, but that doesn't mean nobody is allowed to help me on the rare occasion when I ask for help required.
So you are trying to do something beyond your abilities
Originally posted by @humyI have just made some new applied-math discoveries that renders that above formula almost obsolete so no longer quite so concerned with it as I was.
I have totally given up on simplifying those expressions but now got one more similar one which I hope can somehow simplify;
( ∑[k=c, ∞] (k + m – 1)! / ( k^2 (k + 1) (k + 2m – 1)! ) ) / ∑[k=c, ∞] (k + m – 1)! / ( k (k + 1) (k + 2m – 1)! )
c, k, m ∈ ℕ
I have tried and failed to simplify it and my feeling is that, despite such ...[text shortened]... ator and the denominator, it probably doesn't but still hope there's something I've missed here.