Crossings and embracings of set-partitions and \(q\)-analogues of the logarithm of the Fourier transform
\textit{I. M. Gessel} [Discrete Math. 40, 69-80 (1982; Zbl 0485.05004)] described a \(q\)-analogue of the exponential formula and \textit{A. Nica} [Commun. Math. Phys. 168, No. 1, 187-207 (1995; Zbl 0818.60096)] studied the \(q\)-analogue of the convolution of probability distributions. Here we see the relation between these two ideas. The \(q\)-analogue of the logarithm of the Fourier transform parallels the \(q\)-analogue of the exponential formula of Gessel. Particular cases along the way turn out to be two \(q\)-versions of the Hermite polynomials (the \(q\)-continuous and \(q\)-discrete polynomials). The methods used, beyong the standard partition arguments and path-counting techniques, include infinite matrices. The technical details can be subtle and intricate.
scientific article; zbMATH DE number 935385
scientific article