I recently stumbled upon the following identity, valid for any real numbers $\alpha\_1,\dots,\alpha\_n$ and $\lambda\_{n1} \leq \dots \leq \lambda\_{nn}$:


$$ \mathrm{det}( e^{\alpha\_i \lambda\_{nj}} )\_{1 \leq i,j \leq n} = V(\alpha) \int\_{GT\_\lambda} \exp( \sum\_{i=1}^n \sum\_{j=1}^i \lambda\_{ij} (\alpha\_{n+1-i}-\alpha\_{n-i}))$$
where $V(\alpha)$ is the Vandermonde determinant
$$ V(\alpha) := \prod\_{1 \leq i < j \leq n} (\alpha\_j - \alpha\_i),$$
$GT\_\lambda$ is the Gelfand-Tsetlin polytope of tuples $(\lambda\_{ij})\_{1 \leq j \leq i < n}$ obeying the interlacing relations $\lambda\_{i+1,j} \leq \lambda\_{i,j} \leq \lambda\_{i,j+1}$ and with the usual Lebesgue measure, and one has the convention $\alpha\_0 := 0$. Thus for instance when $n=1$ one has
$$ e^{\alpha\_1 \lambda\_{11}} = \exp( \lambda\_{11} \alpha\_1 )$$
when $n=2$ one has
$$ \mathrm{det} \begin{pmatrix} e^{\alpha\_1 \lambda\_{21}} & e^{\alpha\_1 \lambda\_{22}} \\ e^{\alpha\_2 \lambda\_{21}} & e^{\alpha\_2 \lambda\_{22}} \end{pmatrix} $$
$$= (\alpha\_2 - \alpha\_1) \int\_{\lambda\_{21} \leq \lambda\_{11} \leq \lambda\_{22}} \exp( \lambda\_{11} (\alpha\_2-\alpha\_1) + \lambda\_{21} \alpha\_1 + \lambda\_{22} \alpha\_1 )\ d\lambda\_{11}$$
and when $n=3$ one has
$$ \mathrm{det} \begin{pmatrix} e^{\alpha\_1 \lambda\_{31}} & e^{\alpha\_1 \lambda\_{32}} & e^{\alpha\_1 \lambda\_{33}} \\ e^{\alpha\_2 \lambda\_{31}} & e^{\alpha\_2 \lambda\_{32}} & e^{\alpha\_2 \lambda\_{33}} \\ e^{\alpha\_3 \lambda\_{31}} & e^{\alpha\_3 \lambda\_{32}} & e^{\alpha\_3 \lambda\_{33}} \end{pmatrix} $$
$$ = (\alpha\_2 - \alpha\_1) (\alpha\_3 - \alpha\_1) (\alpha\_3 - \alpha\_2) \int\_{\lambda\_{31} \leq \lambda\_{21} \leq \lambda\_{32}} \int\_{\lambda\_{32} \leq \lambda\_{22} \leq \lambda\_{33}} \int\_{\lambda\_{21} \leq \lambda\_{11} \leq \lambda\_{22}}$$
$$ \exp( \lambda\_{11} (\alpha\_3-\alpha\_2) + \lambda\_{21} (\alpha\_2-\alpha\_1) + \lambda\_{22} (\alpha\_2-\alpha\_1) + \lambda\_{31} \alpha\_1 + \lambda\_{32} \alpha\_1 + \lambda\_{33} \alpha\_1)$$
$$ d \lambda\_{11} d\lambda\_{22} d\lambda\_{21},$$
and so forth.


The identity can be proven easily by induction. I first discovered it by starting with the Schur polynomial identity
$$ \mathrm{det}( x\_j^{a\_i} )\_{1 \leq i,j \leq n} = V(x) \sum\_T x^{|T|}$$
where $0 \leq a\_1 < \dots < a\_n$ are natural numbers in increasing order, $T$ ranges over column-strict Young tableaux of shape $a\_n-n+1, \dots, a\_2-1, a\_1$ with entries in $1,\dots,n$, and $x^{|T|} := x\_1^{c\_1} \dots x\_n^{c\_n}$ where $c\_i$ is the number of occurrences of $i$ in $T$, and taking a suitable "continuum limit" as the $a\_i$ go to infinity and the $x\_j$ go to one in a particular fashion. It can also be derived from the [Duistermaat-Heckmann formula](https://en.wikipedia.org/wiki/Duistermaat%E2%80%93Heckman_formula) for the Fourier transform of [Schur-Horn measure](https://en.wikipedia.org/wiki/Schur%E2%80%93Horn_theorem), combined with the fact that this measure is the pushforward of Lebesgue measure on the Gelfand-Tsetlin polytope under a certain linear map.


Note that the identity also provides an immediate proof that any $n$ distinct exponential functions on $n$ distinct real numbers are linearly independent.


I am certain that this formula already appears in the literature, and perhaps even has a standard name, but I was unable to locate it with standard searches. So my question here is if anyone recognizes the formula and can supply a reference for it.



Reference for exponential Vandermonde determinant identity
This looks like a special case of a formula by Samson Shatashvili related to the HCIZ integral as mentioned in Ryan's answer. Compare, in particular the two ways of computing $\langle 1\rangle$ given by Equations 3.2 and 3.4 in ["Correlation Functions in The Itzykson-Zuber Model"](https://arxiv.org/abs/hep-th/9209083) (thanks to Leonid Petrov for letting me know about this reference in his answer to this [MO question](https://mathoverflow.net/questions/256066/formula-for-un-integration-wanted)). 


