A higher-order generalization of Jacobi’s derivative formula and its algebraic geometric analogue

We generalize Jacobi’s derivative formula for odd m by writing an m×m determinant of higher order derivatives at 0 theta functions in 1 variable with characteristic vectors entries 2mℤ as explicit constant times a power Dedekind’s η-function. do so deriving it from algebraic geometric version that holds not dividing 6m.