In this paper the following generating functions: \[ \exp\left[{x \over 2}\left(t- {1\over t}\right)+ {y\over 2}\left(t^2\tau- {1\over t^2\tau} \right)\right]= \sum^\infty_{n= -\infty}t^nJ_n(x,y; \tau)\tag{*} \]
\[ \exp\left\{ {x\over 2}\left[\left( u-{1\over u}\right)+\left( v-{1\over v}\right) +\left(uv-{1\over uv}\right) \right]\right\} =\sum^\infty_{n= -\infty} \sum^\infty_{m=-\infty} u^mv^nJ_{m,n}(x) \tag{**} \] are considered [see G. Dattoli, A. Torre, S. Lorenzutta and G. Maino, Ann. Numer. Math. 2, No. 1-4, 211-232 (1995; Zbl 0851.33001)], where \(J_n(x,y;\tau)\) is the two-variable one-parameter generalized cylinder function and \(J_{m,n}(x)\) is the double-index Bessel function introduced by P. Humbert through the expansion (**) [Acta Pont. Accad. Sci. Novi Lyncaei 87, 323-331 (1934; Zbl 0009.11503)]. Making use of (*) as well as the two-variable generalized Hermite polynomials \(H_n(x, y)\) defined by the generating function \(\exp(xt+yt^2)\) (loc. cit.) the authors obtain a multiplication theorem for \(J_n(\lambda x,\mu y;\tau)\). Moreover, starting from two suitable different expressions of (**) and using the Hermite-type polynomials \(H_{r,s}(x)\) defined by certain series (loc. cit.) they also establish two other multiplication theorems for the double-index Bessel function \(J_{m,n}(\lambda x)\).