In this paper, I'll be presenting and proving a new identity which I dub as "". I happened upon it while investigating on the Binomial Coefficients in the denominator. It proves great in finding infinite sums where binomial coefficients appear in the denominator. Also, a special thanks to Cody Johnson for helping me out to simplify the identity and coming up with another interesting result from this identity. Also, shoutout to a Brilliant user, Jon Haussmann, who came up with an another amazing proof for this identity.
We check if the following limit exists and results in a finite value where
Thus, we can conclude via the ratio test that the series converges.
We observe the denominator closely. where denotes the Beta function. Thus, we can express the sum as Now, since , we can use the Geometric Expansion that is , to condense the sum as The only work left is to evaluate the integral And thus,
Now, we try telescoping the following series : And thus,
The special case when , the double sum condenses to a single one and we get the following result
We proceed in a similar manner as we did earlier.
We write the denominator as Now, we can express the sum as The only work left is to evaluate the integral.
Let, And thus,
And therefore, the result follows.
Kishlaya's Identity can be generalized for an even number of variables as follows
For odd number of variables, the result follows as
Setting in the Kishlaya's Identity yields
Prove for all
We first try to simplify the sum on the R.H.S. Observe that
After canceling out the common factor (i.e. ) on both sides, it suffices to prove that
Now, we observe that the in the numerator on L.H.S is the derivative of which gives us a clue to differentiate the Kishlaya's Identity w.r.t and we get
Prove Lehmer's Identity, for all
We make the substitution in special case of Kishlaya's Identity and the result follows.
Cody Johnson, came up with an interesting observation related to Kishlaya's Identity.
We again call Beta Function for help.
And thus, once again we thank the Beta Function.
We make use of the MacLaurin Expansion of On comparing the coefficients, we can conclude the following