1
$\begingroup$

About non central chi-square, I have read paper from Castano-Martinez and Lopez-Blazquez(2005). They made expansion of $\chi_n^2(\delta)$ using inverse-Laplace transform and used Laguerre polynomial. I have a little confusion about its notation.

In page 4, equation 2.3, the probability density function of $ \chi_n^2(\delta)$ is:

$f(y)=\frac{e^{-y/2}}{2^{n/2}}\frac{y^{n/2-1}}{\Gamma(n/2)}\sum_{k=0}^{\infty}\frac{(-\delta/2)^k}{(n/2)_k}L_k^{(n/2-1)}(\frac{y}{2}) $

The one that confuses me is subscript $k$ in denominator expression $(n/2)_k$ (after $\sum$). What does $k$ means? is it a power or index?

$\endgroup$
1
$\begingroup$

The subscript $k$ of $(n/2)_k$ implies notation for the Pochhammer. Loosely put, a Pochhammer is an incrementing factorial, as opposed to a more typical decrementing factorial.

More exactly, factorial $n$, $n!=n(n-1)(n-2)...(3 )(2) (1)$ for positive integers generalizes to $\Gamma(n+1)$, for $n$ is almost any real number with the exception of negative integers, which are discontinuities, where $\Gamma(.)$ is the gamma function.

Similarly, the Pochhammer $(n)_k=n(n+1)(n+2)...(n+k-2)(n+k-1)$ for positive integers generalizes to $(n)_k=\frac{\Gamma(n+k)}{\Gamma(n)}$ for $n$ is any real number (negative integers as well). Obviously, the negative integer Pochhammer products are obtained by simple integer multiplications and not obtained using the $\Gamma$ identity.

Most people have never heard of a Pochhammer. Thus, having trekked through this post, you know something mathematical that most people do not. Congratulations!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.