Abstract
The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.
Highlights
Methods employing the concepts and the formalism of umbral calculus have been exploited in [1]to guess the existence of generating functions involving harmonic numbers [2]
The conjectures put forward in [1] have been proven in [3,4] and further elaborated in [5], and these were extended to hyper-harmonic numbers in [6]
We use the same point of view as [1], by discussing the possibility of exploiting the formalism developed therein in a wider context
Summary
Methods employing the concepts and the formalism of umbral calculus have been exploited in [1]. To guess the existence of generating functions involving harmonic numbers [2]. We use the same point of view as [1], by discussing the possibility of exploiting the formalism developed therein in a wider context. The key idea is that of exploiting the harmonic number index as a power exponent; such a “promotion” allows the possibility of reducing the associated computational technicalities to elementary algebraic manipulations. Series involving harmonic numbers can, e.g., be treated as formal series of known functions (exponential, Gaussian, rational, etc.), and the relevant properties can be exploited to carry out computations, which are significantly more cumbersome and involved when conventional methods are employed
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