An umbral calculus for polynomials characterizing U(n) tensor operators (Q790314)

A simple algorithm, called an ''umbral calculus'', is given to compute the invariant polynomials \(_{\mu}G_ q^{(n)}(\gamma_ 1,...,\gamma_ n;\delta_ 1,...,\delta_ n)\) characterizing certain U(n) tensor operators. It is shown that these polynomials can be written as integral linear combinations of Schur functions \(S_{\lambda}(\lambda -\delta)\) in the symbol \(\gamma\)-\(\delta\), where \(\gamma\)-\(\delta\) denotes the difference of the two sets of variables \(\{\gamma_ 1,...,\gamma_ n\}\) and \(\{\delta_ 1,...,\delta_ n\}\). A similar result is obtained for the more general bisymmetric polynomials \(^ m_{\mu}G_ q^{(n)}(\gamma_ 1,...,\gamma_ n;\delta_ 1,...,\delta_ m).\) Up to a simple factor the classical well-posed hypergeometric series \(_{j+1}F_ j\) are special cases of the polynomials \(^{j- 2}_{0}G_{q}^{(2)}\) [\textit{W. J. Holman}, the first author and \textit{J. D. Louck}, SIAM J. Math. Anal. 7, 529-541 (1976; Zbl 0329.33013) and \textit{S. C. Milne}, Adv. Math. 36, 169-211 (1980; Zbl 0451.33010)]. One application of the umbral calculus is to show that the ''transposition'' and ''conjugation'' symmetries for the \({}^ m_{\mu}G_ q^{(n)}\) functions are equivalent. [For background see \textit{R. A. Gustafson} and \textit{S. C. Milne}, Adv. Appl. Math. 4, 422-478 (1983; Zbl 0534.33009). A proof of transposition symmetry is given in a forthcoming paper of \textit{R. A. Gustafson} and \textit{S. C. Milne} [A new symmetry for Biedenharn's G-functions and classical hypergeometric series (in preparation)].