We know that there exists similarities between power functions and derivative of a function (in particular, Newton binomial formula and Leibniz rule for derivation of a product can be deduced from each other). Thanks to the alternant formula, we can express the determinant of a Vandermonde matrix with a missing power (e.g. see this post) via a power function of its coefficients.

Question: Does it exist a similar formula for the Wronskian of arbitrary functions :

$$ W_k(f_1,\ldots, f_{n}) = \begin{vmatrix} f_1&\ldots&f_1^{(k-1)}&f_1^{(k+1)}&\ldots &f_1^{(n)}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ f_n&\ldots&f_n^{(k-1)}&f_n^{(k+1)}&\ldots &f_n^{(n)} \end{vmatrix} $$

We know that there exists similarities between power functions and derivative of a function (in particular, Newton binomial formula and Leibniz rule for derivation of a product can be deduced from each other). Thanks to the alternant formula, we can express the determinant of a Vandermonde matrix with a missing power (e.g. see Computing an almost Vandermonde matrix) via a power function of its coefficients.

Question: Does there exist a similar formula for the Wronskian of arbitrary functions :

$$ W_k(f_1,\dotsc, f_{n}) = \begin{vmatrix} f_1&\cdots&f_1^{(k-1)}&f_1^{(k+1)}&\cdots &f_1^{(n)}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ f_n&\cdots&f_n^{(k-1)}&f_n^{(k+1)}&\cdots &f_n^{(n)} \end{vmatrix}? $$