My binomials have recursiveness

My binomials have recursive property.

Here is a Wolfram screenshot to confirm this. The binomial terms on the LHS are a by b whole to the power of x type in both numerator and denominator. The RHS also is a by b whole to the power of k.

You may recall that my binomials if divided for sigma till n by sigma till n-k gives a by b whole to the power of k approximately. Below is what I mean to say 

What I mean is below

And here are screenshots to confirm that my just above statement is true

Have a nice day !!