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Non integral binomial coefficient

Posted by peeterjoot on April 10, 2013

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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

\begin{aligned}f(x) = (a + x)^b.\end{aligned} \hspace{\stretch{1}}(1.1)

Our derivatives are

\begin{aligned}\begin{aligned}f'(x) &= b (a + x)^{b-1} \\ f''(x) &= b(b-1) (a + x)^{b-2} \\ f^3(x) &= b(b-1)(b-(3-1)) (a + x)^{b-3} \\ \dot{v}s & \\ f^k(x) &= b(b-1)\cdots (b-(k-1)) (a + x)^{b-k}.\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.2)

Our Taylor series is then

\begin{aligned}(a + x)^b = \sum_{k = 0}^\infty \frac{1}{{k!}} b(b-1)\cdots (b-(k-1)) a^{b-k} x^k.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Note that if $b$ is a positive integer, then all the elements of this series become zero at $b = k - 1$, or

\begin{aligned}(a + x)^b = \sum_{k = 0}^k \frac{1}{{k!}} b(b-1)\cdots (b-(k-1)) a^{b-k} x^k.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Gamma function

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

\begin{aligned}\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt.\end{aligned} \hspace{\stretch{1}}(1.0.2)

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

\begin{aligned}\Gamma(z + 1) &= \int_0^\infty t^{z} e^{-t} dt \\ &= \int_0^\infty t^{z} d \left( \frac{ e^{-t}}{-1} \right) \\ &= {\left.{{t^z \frac{e^{-t}}{-1}}}\right\vert}_{{0}}^{{\infty}}- \int_0^\infty z t^{z-1} \frac{e^{-t}}{-1} dt \\ &= z \int_0^\infty t^{z-1} e^{-t} dt \\ &= z (z - 1)\int_0^\infty t^{z-2} e^{-t} dt \\ &= z (z - 1)(z - (3-1))\int_0^\infty t^{z-3} e^{-t} dt \\ &= z (z - 1) \cdots (z - (k-1))\int_0^\infty t^{z-k} e^{-t} dt \\ &= z (z - 1) \cdots (z - (k-1))\int_0^\infty t^{(z + 1 - k) - 1} e^{-t} dt,\end{aligned} \hspace{\stretch{1}}(1.0.2)

or

\begin{aligned}\Gamma(z + 1) = z (z - 1) \cdots (z - (k-1)) \Gamma( z - (k-1) ).\end{aligned} \hspace{\stretch{1}}(1.0.2)

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

\begin{aligned}z (z - 1) \cdots (z - k)=\frac{\Gamma(z + 1)}{\Gamma( z - (k-1) )}.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Binomial coefficient for positive exponents

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

\begin{aligned}(a + x)^b &= \sum_{k = 0}^\infty \frac{1}{{k!}} \frac{\Gamma(b+1)}{\Gamma(b - k + 1)}a^{b-k} x^k \\ &= \sum_{k = 0}^\infty \frac{\Gamma(b + 1)}{\Gamma(k + 1)\Gamma(b - k + 1)}a^{b-k} x^k.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Observe that when $b$ is a positive integer we have

\begin{aligned}\frac{\Gamma(b + 1)}{\Gamma(k + 1)\Gamma(b - k + 1)} &= \frac{b!}{k!(b-k)!} \\ &= \binom{b}{k}.\end{aligned} \hspace{\stretch{1}}(1.0.10)

So for positive values of $b$, 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

\begin{aligned}\binom{b}{k} \equiv\frac{\Gamma(b + 1)}{\Gamma(k + 1)\Gamma(b - k + 1)}.\end{aligned} \hspace{\stretch{1}}(1.0.11)

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

\begin{aligned}(a + x)^b = \sum_{k = 0}^\infty \binom{b}{k} a^{b-k} x^k.\end{aligned} \hspace{\stretch{1}}(1.0.11)

Binomial coefficient for negative integer exponents

Using the relation eq. 1.0.11 blindly leads to some trouble, since $\Gamma(-\left\lvert {m} \right\rvert)$ goes to infinity for integer values of $m > 0$. We have to modify the definition of the binomial coefficient. Let’s rewrite eq. 1.0.2 for negative integer values of $b = -m$ as

\begin{aligned}(a + x)^{-m} &= \sum_{k = 0}^\infty \frac{1}{{k!}} (-m)(-m-1)\cdots (-m-(k-1)) a^{-m-k} x^k \\ &= \sum_{k = 0}^\infty \frac{1}{{k!}} (-1)^km(m+1)\cdots (m+(k-1)) a^{-m-k} x^k.\end{aligned} \hspace{\stretch{1}}(1.0.11)

Let’s also put the ratio of gamma functions relation of eq. 1.0.2, in a slightly more general form. For $u, v > 0$, where $u - v$ is an integer, we can write

\begin{aligned}u (u - 1) \cdots (v) = \frac{\Gamma(u + 1)}{\Gamma( v )}.\end{aligned} \hspace{\stretch{1}}(1.0.11)

Our Taylor series takes the form

\begin{aligned}(a + x)^{-m} &= \sum_{k = 0}^\infty \frac{1}{{k!}} (-1)^k(m+k-1) (m+k-2) \cdots (m)a^{-m-k} x^k \\ &= \sum_{k = 0}^\infty (-1)^k \frac{ \Gamma(m + k) }{\Gamma(m)\Gamma(k+1)}a^{-m-k} x^k.\end{aligned} \hspace{\stretch{1}}(1.0.16)

We can now define, for negative integers $-m$

\begin{aligned}\binom{-m}{k}\equiv(-1)^k \frac{ \Gamma(m + k) }{ \Gamma(m)\Gamma(k+1) }.\end{aligned} \hspace{\stretch{1}}(1.0.16)

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

\begin{aligned}(a + x)^{-m} = \sum_{k = 0}^\infty \binom{-m}{k} a^{-m-k} x^k.\end{aligned} \hspace{\stretch{1}}(1.0.16)

For negative integer values of $b = -m$, 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

\begin{aligned}\binom{-m}{k} &= (-1)^k \frac{ \Gamma(m + k) }{ \Gamma(m)\Gamma(k+1) } \\ &= (-1)^k \frac{ (m + k -1)! }{ (m-1)! k! } \\ &= (-1)^k \frac{ m (m + k)! }{ (m + k) m! k! },\end{aligned} \hspace{\stretch{1}}(1.0.16)

or

\begin{aligned}\binom{-m}{k}=(-1)^k \frac{m}{m + k} \binom{m+k}{m}.\end{aligned} \hspace{\stretch{1}}(1.0.19)

Negative non-integral binomial coefficients

TODO. There will be some ugliness due to the changes of sign in the products $b(b-1)\cdots (b -k + 1)$ since $b$ and $b -k + 1$ 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.