\(K_X(t)=\log M_X(t)\)

let \(X\) is a non central \(\chi^2\) variate with parameters degrees of freedom,\(n\) and non-centrality parameter,\(\lambda\).

Moment generating function of \(X\) is:

\( M_X(t)=(1-2t)^{-\frac{n}{2}}\exp(\frac{\lambda t}{1-2t})\)

\( K_X(t)=\log M_X(t)=\log [(1-2t)^{-\frac{n}{2}}\exp(\frac{\lambda t}{1-2t})]\)

\( K_X(t)={-\frac{n}{2}}\log (1-2t)+(\frac{\lambda t}{1-2t})\)

\( K_X(t)={-\frac{n}{2}}[-2t-\frac{(2t)^2}{2}-\frac{(2t)^3}{3}-\frac{(2t)^4}{4}-\ldots]+(\frac{\lambda t}{1-2t})\ldots\ldots (A)\)

\(r^{th}\) cumulant is the coefficient of \(\frac{t^r}{r!}\) in \(K_X(t)\).

But i couldn't separate \(\frac{t^r}{r!}\) in \(K_X(t)\) because of the denominator \((1-2t)\).

Hence couldn't able to compute the cumulants to know the mean,variance, 3rd&4th central moment of the distribution.

Can you please rearrange the equation \((A)\) as \(\frac{t^r}{r!}\) so that the four cumulants can be obtained easily.