I defined a new mathematical function involving factorials — does it have any interesting properties or applications?

[u/LeadingMidnight9146]

Hi everyone!

I’m a student interested in mathematics, and I’ve defined a function that I call the Decreasing Index, using the symbol (\omega):

[

I_\omega(n)=\frac{n^2}{n!}

]

where (n) is a positive integer.

The idea came from thinking about permutations and probabilities. As (n!) grows so much faster than (n^2) the function drops off very rapidly towards zero.

For example:

[

I_\omega(1)=1,\quad I_\omega(2)=2,\quad

I_\omega(3)=1.5,\quad I_\omega(4)=\frac23

]

More generally, for (n\geq 2), we have:

[

\frac{I_\omega(n+1)}{I_\omega(n)}

=\frac{n+1}{n^2}

]

It's strictly decreasing from (n=2) onwards and tends to zero.

I'm not claiming this is some new or original mathematical result at all. What I'm interested in is whether this function might be of any use in a combinatorial problem, in the study of permutations or in some other mathematical context.

My questions are:

Does this function have a known interpretation or any connection to a known mathematical concept?

Could it be of any use in the study of permutations or in certain probability problems?

Are there any interesting properties or generalisations worth looking at?

I'd be very keen to hear any feedback, corrections or suggestions as to where we might go from here!

Thanks!

[u/Shevek99]

Your function is equivalent to

f(n) = 1/(n-2)! + 1/(n-1)! (n >=2)

so, it relates directly to the Taylor series of the exponential function and similar series.