If is a positive integer, the expansion of has each term being a product of variables of the form where ranges from 0 to n. The coefficient of is precisely the number of ways we can choose of the variables to be , which is . Hence we have the binomial theorem:
But what if is not a positive integer? This post is about how we extend this formula and why it still holds. Interestingly the same formula holds with minor modifications. Firstly we change the upper limit of the sum to infinity – the sum will then only converge in particular circumstances. Secondly we extend the definition of the binomial coefficient to complex values of via .
Note that if is a complex number then is defined for any non-negative integer – we simply do repeated multiplication, and define (even for ). However a non-integer power of a complex number is not straightforward to define unless the number is a positive real. If we can define where is the unique real solution to . Extending the definition to other complex numbers leads to issues with multifunctions or discontinuities (e.g. could be one of two values or ). Hence we are going to restrict ourselves to the case and real.
While the term is fine, we need to take care with if is no longer an integer. Hence we shall restrict ourselves to non-negative . The binomial theorem is valid when so consider . When does the infinite series converge? By the ratio test, the sum converges if the limit of converges to a number less than 1 as . (The sum does not converge if the limit is greater than 1 and if the limit either does not converge or is equal to 1, then the test is inconclusive.) Applying this test to our case of gives us
This has limit less than 1 as provided . We can thus state the following.
If is a complex number, is real, is positive and , then the sum converges.
The reason this sum is equal to is a consequence of the Taylor series expansion of about and the fact that the identity is valid when . Also note that the condition implies so is well defined.
We can also use a differential equations approach to proving this. Fix and consider as a power series in valid for . The power series is differentiable term by term within this interval of convergence and so
We may also write this as
Adding times (2) to times (3),
Hence by (4)
so we deduce that is constant. For this is , and we conclude that . Summarising, we have the following result.
If is a complex number, is real, is positive and , then
A particularly attractive special case of this formula is for and replaced with :
- Newton’s generalised binomial theorem – Wikipedia
- Binomial Theorem/General Binomial Theorem – ProofWiki