Higher dimensional recursive matrices and diagonal delta sets of series

Abstract

The idea to study infinite matrices whose entries are the coefficients of the powers of a given formal series is rather old and dates back at least to Schur’s posthumous papers on Faber polynomials [39-41]. In 1953, Jabotinsky reconsidered Schur and Shiffer’s [38] work on the subject and developed a systematic study of these matrices [20]. Since then, several applications confirmed that Jabotinsky matrices are a useful tool in computational analysis and in the theory of formal series (see, e.g., [ 16, 17, 21, 251). However, there exists a totally different motivation in order to undertake this kind of study. Not long ago, moving from considerations on the umbra1 calculus-the operator calculus associated with polynomial sequences of Sheffer type [ 1, 13, 28, 32, 36, 37]-the authors of [l] were led to introduce a class of matrices which turns out to be an extension of that studied by Jabotinsky. These matrices, called “recursive matrices,” can be cogently interpreted as matrices of integer evaluations of polynomial sequences of Sheffer type as well as matrices representing the groups of umbra1 and shift- invariant operators, respectively. A self-contained theory of recursive matrices has been worked out in [2] and provides generalized versions of Jabotinsky’s results; on the other hand, it can be used (see, e.g., [3]) to extend and simplify the umbra1 calculus and some related topics as, for example, the theory of binornial functions [29-321. The purpose of the present paper is to contribute to the higher-dimensional reformulation of [2]. This program is to be at least twofold since it has to reflect the fact that, in the multivariate case, two different definitions are available for the algebra of Laurent series. 315