It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.

But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code for computing Hypergeometric functions of matrix argument. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:

\begin{equation*} {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx, \end{equation*} where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.

(PS: This integral has previously been discussed on MO, e.g., here in this question of L. Nicolaescu)

Edit: Regarding numerical approximation. Here is what seems to be the latest in this direction. Have a look at the geodesic monte carlo sampling method of Byrne and Girolami; (they discuss sampling from the Bingham distribution). Once you have that, then it should be "easy" to estimate the normalization constant. But I guess the bad thing might again be a lack of guarantees on how long it takes to get a given accuracy approximation---but for now, seems like the abovecited approach might be the most promising.

It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.

But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code for computing Hypergeometric functions of matrix argument. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:

\begin{equation*} {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx, \end{equation*} where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.

(PS: This integral has previously been discussed on MO, e.g., here in this question of L. Nicolaescu)

Edit: Regarding numerical approximation. Here is what seems to be the latest in this direction. Have a look at the geodesic monte carlo sampling method of Byrne and Girolami; (they discuss sampling from the Bingham distribution). Once you have that, then it should be "easy" to estimate the normalization constant. But I guess the bad thing might again be a lack of guarantees on how long it takes to get a given accuracy approximation---but for now, seems like the abovecited approach might be the most promising.

It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.

But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code for computing Hypergeometric functions of matrix argument. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:

\begin{equation*} {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx, \end{equation*} where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.

(PS: This integral has previously been discussed on MO, e.g., here in this question of L. Nicolaescu)

It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.

But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code for computing Hypergeometric functions of matrix argument. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:

\begin{equation*} {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx, \end{equation*} where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.

(PS: This integral has previously been discussed on MO, e.g., here in this question of L. Nicolaescu)

Edit: Regarding numerical approximation. Here is what seems to be the latest in this direction. Have a look at the geodesic monte carlo sampling method of Byrne and Girolami; (they discuss sampling from the Bingham distribution). Once you have that, then it should be "easy" to estimate the normalization constant. But I guess the bad thing might again be a lack of guarantees on how long it takes to get a given accuracy approximation---but for now, seems like the abovecited approach might be the most promising.