This integral is needed to obtain the joint distribution of the sample variances of a random sample from a bivariate Gaussian distribution. For details on the joint distribution of the sample means, sample variances, and sample correlation, please see Sections 14.11 to 14.13 of Kendall's book (2nd edition):
Kendall, Maurice G., The Advanced Theory of Statistics. Vol. I, Philadelphia: J. B. Lippincott Co., XI, 457 p. (1944). ZBL0063.03214.
The integral is $$ I\left( n,c\right) =\int_{-1}^{1}\exp\left( cr\right) \left( 1-r^{2}\right) ^{\left( n-4\right) /2}dr,$$
where $c\neq0$ is a constant, and $n\geq4$ is (a constant and) an integer (and is the sample size). When $n-4$ is even, we can use ``integration by parts'' to get $I\left( n,c\right) $ after some tedious computations. In general, we can use the expansion $$\left( 1-r^{2}\right) ^{\left( n-4\right) /2}=\sum_{k=0}^{\infty}a_{k,n}r^{2k},$$ compute each $$s_{k}=\int_{-1}^{1}\exp\left( cr\right) a_{k,n}r^{2k}dr,$$ and then compute $\sum_{k=0}^{\infty}s_{k}$. For this method, we may not have an explicit, analytic expression for $I\left( n,c\right) $.
On the other hand, we can use the expansion $$\exp\left( cr\right) =\sum_{k=0}^{\infty}\frac{c^{k}r^{k}}{k!},$$ compute $$b_{2k}=2\int_{0}^{1}% \frac{c^{2k}}{\left( 2k\right) !}r^{2k}\left( 1-r^{2}\right) ^{\left( n-4\right) /2}dr,$$ and compute $\sum_{k=0}^{\infty}b_{2k}$. Note that each $b_{2k}$ relates to the Beta function (upon a change of variable $r^{2}% \mapsto\tau$). But what does $\sum_{k=0}^{\infty}b_{2k}$ look like?
Could someone please point me to some references on whether $I\left( n,c\right) $ has an explicit, analytic expression, or on whether such an expression can be obtained by one of the attempts mentioned above? Dose this integral involve special functions? Thanks.
