Algebraic proof of the raising-operator identity

Prove a general algebraic identity establishing that the raising-operator formula \(s_{\lambda}(p\mid\alpha)=\prod_{1\leqslant i<j\leqslant\ell}(1-R_{ij})h_{\lambda,\delta}\) agrees with the corresponding integral formula for double Schur functions, including the specialization \(\beta=0\).

Background

The paper compares an integral formula for double Schur functions with the raising-operator formula introduced in Fun's work. Although the specialization with the second parameter set equal to zero reduces to the classical integral-formula argument, the authors observe that the expected extension to the general parameter setting does not follow straightforwardly.

The unresolved task is specifically a general algebraic proof of the identity illustrated in Example \ref{ex:raising_operator}, rather than merely a future direction or a comparison of formulas.