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**imcute****Member**- Registered: 2022-09-28
- Posts: 176

a double factorial products up 2 at a time not 1

2x!!=product k=1->x 2x

so x+2n!!=x!!*product k=1->n 2x

for something like a gamma function,we can take the log

L(x)=ln(x!!)

N is a huge number

so ln(N+k)-ln(N)=0 for a little k compared to N

L(N+2n)=L(N)+sum k=1->n ln(N+2k)

=L(N)+sum k=1->n ln(N)

=L(N)+nlnN

now do some substitution

L(2n)+sum k=1->N/2 ln(2n+2k)=sum k=1->N/2 ln(2k)+nlnN

move it to right and merge sums

L(2n)=sum k=1->N/2 -ln(2n+2k)+ ln(2k)+nlnN

log rule

L(2n)=sum k=1->N/2 ln(k/k+n)+nlnN

sub n for 2n and exponentiate

n!!=product k=1->N/2 k/k+n/2 +N^(n/2)

minor changes

n!!=product k=1->N/2 2k/k+n +sqrt(N^n)

done!

change the limits a bit N->2N^2 x->n

x!!=lim N->inf 4*N^n*product k=1->N^2 k/(x+2k)

idk if it actually works but i think it would cuz it based on the factorial video from lines that connect

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 41,308

Hi iccute,

Here is a relevant link: Double Factorial.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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