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  1. ; ; ; ; (, Forum of Mathematics, Sigma)
    Abstract We give an explicit raising operator formula for the modified Macdonald polynomials$$\tilde {H}_{\mu }(X;q,t)$$, which follows from our recent formula for$$\nabla $$on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions$$\tilde {H}^{1,n}(X;q,t)$$that we call$$1,n$$-Macdonald polynomials, which reduce to a scalar multiple of$$\tilde {H}_{\mu }(X;q,t)$$when$$n=1$$. We conjecture that the coefficients of$$1,n$$-Macdonald polynomials in terms of Schur functions belong to$${\mathbb N}[q,t]$$, generalizing Macdonald positivity. 
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    Free, publicly-accessible full text available January 1, 2026
  2. ; ; ; ; (, Journal für die reine und angewandte Mathematik)
    Abstract. We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies in the algebra of symmetric functions embedded in the elliptic Hall algebra of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the nabla operator applied to any LLT polynomial. In particular, we obtain a formula for ∇msλ which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion paper to this one 
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    Accepted Manuscript Full Text Available
  3. ; ; ; ; (, Forum of Mathematics, Pi)
    Abstract We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $$\operatorname {\mathrm {GL}}_{l}$$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials. 
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  4. ; ; ; ; (, Forum of Mathematics, Pi)
    Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $$\Delta _{h_l}\Delta ' _{e_k} e_{n}$$ , where $$\Delta ' _{e_k}$$ and $$\Delta _{h_l}$$ are Macdonald eigenoperators and $$e_n$$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $$\operatorname {\mathrm {GL}}_m$$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions. 
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