Gamma functions

 The three consecutive numbers and four consecutive numbers fetch beautiful gamma functions. 

Here are the Wolfram screenshots

You may recall that gamma functions are nothing but factorials.

Gamma function of n is (n-1)!

So essentially the first result is (2m+1)!. So the product of three consecutive numbers (in the above fashion ) taken in odds / or evens (depending on how one looks) give factorials.

It is interesting if we divide the results obtained above. What we get is an expression of pi. See the Wolfram result below:-

What I have done above is the division of the first result with the second result. Essentially the product of the three consecutive numbers and the product of four consecutive numbers have been divided. In the first two cases they were considered till m. In the third till infinity. We get an expression of pi.

This is the second expression for pi with 3 consecutive numbers and 4 consecutive numbers.

Hope you like these expressions. Have a fine day!!