# Expansion of Non-central Chi-square density function

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?

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.