A unified point of view on the theory of generalized Bessel functions
Continuing the treatment of certain generalized Bessel functions (G.B.F.) studied in a number of previous works, the authors in the paper under review discuss several properties of a two-variable G.B.F. denotes \({\mathcal D}_n^{(p, m)} (x, y)\) introduced in another earlier paper [see \textit{G. Dattoli}, \textit{S. Lorenzutta}, \textit{G. Maino} and \textit{A. Torre}, Ann. Numer. Math. 2, 211-232 (1995; Zbl 0851.33002)]. Most of these properties (recurrence relations, generating functions, Graf type addition theorem, \dots) as well as some results of the particular cases \({\mathcal D}_n^{(1, m)} (1, y)= W^m_n (y)\) and \({\mathcal D}_n^{(1,1)} (1,y)= C_n(y)\) where \(W^m_n\) and \(C_n\) denote the Wright and Bessel-Clifford functions, respectively, were already given in the mentioned paper. This \({\mathcal D}_n^{(p, m)} (x,y)\) is here viewed as a two-variable function of the type introduced by \textit{W. Miller} [Commun. Pure Appl. Math. 18, 493-499 (1965; Zbl 0133.32201)] and its link with the two-variable generalized Hermite polynomials \(H_n^{(p, m)} (x, y)\) of \textit{H. W. Gould} and \textit{A. T. Hopper} [Duke Math. J. 29, 51-63 (1962; Zbl 0108.06504)] is also stated. Finally, a few comments related to the Lie algebraic structure of \({\mathcal D}_n^{(p, m)} (x, y)\) along with some graphs for particular values of their indices, are given. \{Reviewer's remarks: For a more complete information on the Bessel-Clifford functions, it is recommended to see, e.g., the works by the reviewer [Collect. Math. 18 (1966-1967), 57-174, Barcelona (1967; Zbl 0153.40301) and Tables of integrable differential equations by Bessel-Clifford functions (Spanish), Ser. Inf. 27 Univ. La Laguna, Seret. Publ. 95 p. (1989; Zbl 0717.33001); see specially the list of references of other works by the reviewer on these functions therein]\}.
scientific article; zbMATH DE number 811393