## Non integral binomial coefficient

Posted by peeterjoot on April 10, 2013

Updated. Original generalization of binomial coefficient wasn’t correct for negative exponents.

In [2] appendix section F was the use of binomial coefficients in a non-integral binomial expansion. This surprised me, since I’d never seen that before. However, on reflection, this is a very sensible notation, provided the binomial coefficients are defined in terms of the gamma function. Let’s explore this little detail explicitly.

**Taylor series**

We start with a Taylor expansion of

Our derivatives are

Our Taylor series is then

Note that if is a positive integer, then all the elements of this series become zero at , or

**Gamma function**

Let’s now relate this to the gamma function. From [1] section 6.1.1 we have

Iteratively integrating by parts, we find the usual relation between gamma functions of integral separation

or

Flipping this gives us a nice closed form expression for the products of a number of positive unit separated values

**Binomial coefficient for positive exponents**

Considering first positive exponents , we can now use this in our Taylor expansion eq. 1.0.2

Observe that when is a positive integer we have

So for positive values of , even non-integer values, we see that is then very reasonable to define the binomial coefficient \index{binomial coefficient} explicitly in terms of the gamma function

If we do that, then the binomial expansion for non-integral values of is simply

**Binomial coefficient for negative integer exponents**

Using the relation eq. 1.0.11 blindly leads to some trouble, since goes to infinity for integer values of . We have to modify the definition of the binomial coefficient. Let’s rewrite eq. 1.0.2 for negative integer values of as

Let’s also put the ratio of gamma functions relation of eq. 1.0.2, in a slightly more general form. For , where is an integer, we can write

Our Taylor series takes the form

We can now define, for negative integers

With such a definition, our Taylor series takes the tidy form

For negative integer values of , this is now consistent with eq. 1.0.11.

Observe that we can put eq. 1.0.16 into the standard binomial form with a bit of manipulation

or

**Negative non-integral binomial coefficients**

TODO. There will be some ugliness due to the changes of sign in the products since and may have different sign. A product of two ratios of gamma functions will be required to express this product, which will further complicate the definition of binomial coefficient.

# References

[1] M. Abramowitz and I.A. Stegun. \emph{Handbook of mathematical functions with formulas, graphs, and mathematical tables}, volume 55. Dover publications, 1964.

[2] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

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