Lapointe–Vinet's creation-operator formula for Macdonald polynomials

Let $N$ be the number of variables, let $\lambda=(\lambda_1,\dots,\lambda_N)$ be a partition, and let $J_\lambda(x;q,t)$ denote the Macdonald polynomial. For each $k$-element subset $I$ of $\{1,\dots,N\}$, define

$$x_I=\prod_{i\in I}x_i,$$ $$\tilde A_I(x;t)=t^{-(N-k)k}\prod_{\substack{i\in I\\ j\notin I}}\frac{t x_i-x_j}{x_i-x_j},$$

and let $M_I(X;q,t)=M_k(X;q,t)$ in the $k$ variables $x_i$ with $i\in I$. Set

$$\tilde B_k^+=\sum_I\tilde A_I(x;t)x_I M_I(-t;q,t),$$

where the sum is over all such subsets $I$.

Lapointe–Vinet's creation-operator conjecture. The Macdonald polynomials are given by

$$J_\lambda(x;q,t)=(\tilde B_N^+)^{\lambda_N}(\tilde B_{N-1}^+)^{\lambda_{N-1}-\lambda_N}\cdots(\tilde B_1^+)^{\lambda_1-\lambda_2}\cdot1.$$

This Rodrigues-type formula would imply that the coefficients of the Macdonald polynomials in the monomial basis are polynomials in $q$ and $t$ with integer coefficients, extending analogous formulas for Jack polynomials; its status is not resolved in the supplied text.