About the sum rules of special functions

Generalized Zernike or disc polynomials: An application in quantum optics

“…Indeed, as earlier mentioned, in [6] sums of the type with t being a complex parameter, h ≥ m and r , r ≤ 1, have been proved to be amenable to the closed form expression:…”

“…Functions of the type (4.5) have been considered in [6], where the possibility of giving closed form expressions to sums of products of special functions, and so in particular to sums of products of functions like (4.5), was investigated. According to the discussion in Section 3, it becomes evident that such sums are of some relevance, for instance, in connection with the photon statistics of two-level systems.…”

“…In addition, we note that, apart from the factors (−1) m √ π or (−1) n √ π , functions of the type R m,n (r ) have been considered in [6], where what we may consider the h → ∞ limit of the sum (4.39), namely s m (r, r ; t) = ∞ l=−m t l R m,m+l (r )R m,m+l (r ), (5.14) was proved to have the closed form expression…”

Mittag-Leffler-Gould-Hopper polynomials: Symbolic Approach

“…The multi-variable special polynomials provide the solutions of a wide class of partial differential equations often encountered in the field of physical problems. The importance of multi-variable Hermite polynomials has been recognized in dealing with quantum mechanical and optical beam transport problems [8,14,15]. It happens very often that the solution of a given problem in physics or applied mathematics requires the evaluation of infinite sums involving special functions.…”

“…It has been shown that the summation formulae of certain special functions, often encountered in applications ranging from electromagnetic processes to combinatorics, can be written in terms of the multi-variable Hermite polynomials. [8]. Throughout this paper, we use the notations: N = {1, 2, 3...} and N 0 := N ∪ {0}.…”