Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product

In a preceding article, we introduced stable symmetric series encoding simultaneously the normalized conjugacy classes of all symmetric groups. The same rational series remain stable for the Jack-deformed Goulden–Jackson product. We investigate their two-parameter Macdonald analogue. Starting from the $(q,t)$-deformed class product (dual to the coproduct diagonal on the $J$-basis), we construct the unique infinite series whose multiplication realizes any shifted Macdonald eigenvalue. In contrast with the classical and Jack cases, this transform is no longer multiplication by a fixed explicit series. We identify it with a composition of a Cauchy multiplication, the integral nabla operator, and a simple diagonal operator. We then determine the series realizing the Nazarov–Sklyanin operators $A^{(k)}$, derive a single generating series for all column partitions, and compare our construction with the Macdonald characters and Theta operators of Ben Dali and D’Adderio.