Umbral Methods and Harmonic Numbers

The theory of harmonic based function 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.


Introduction
Methods employing the concepts and the formalism of umbral calculus have been exploited in [1] to conjecture the existence of generating functions involving Harmonic Numbers [2]. The conjectures put forward in [1] have been proven in [3]- [4], further elaborated in subsequent papers [5] and generalized to Hyper-Harmonic Numbers in [6]. In this note we use the same point of view of [1] , by discussing the possibility of exploiting the formalism developed therein in a wider context.
We accordingly remind that harmonic numbers are defined as It is furthermore evident that the integral representation for this family of numbers can be derived using a standard procedure, we first note that ∞ 0 e −s r ds (2) thereby getting [7] after interchanging summation and integral signs. The definition in eq. (3) can be extended to non-integer values of n and, therefore, it can be exploited as an alternative definition holding for n (not-necessarily) a positive real.
We define now the umbral operatorĥ n = h n , with the propertyĥ nĥm =ĥ n+m (5) and introduce the Harmonic Based Exponential Function (HBEF) which, as already discussed in [1], has quite remarkable properties.
The relevant derivatives can accordingly be expressed as (see the concluding part of the paper for further comments) We use the previous definition to derive the following integral It is evident that by expanding the umbral function on the r h s of eq. (8), we obtain an expected conclusion, achievable by direct integration, underscored here to stress the consistency of the procedure.
A further interesting example comes from the following "Gaussian" integral The last term in eq. (10) has been obtained by treatingĥ as an ordinary algebraic quantity and then by applying the standard rules of the Gaussian integration Let us now consider the following slightly more elaborated example, involving the integration of two "Gaussians", namely the ordinary case and its HBEF analogous This last result, obtained after applying elementary rules, can be worded as it follows: the integral in eq. (12) depends on the operator function on its r.h.s., for which we should provide a computational meaning. The use of the Newton binomial yields and the correctness of this conclusion has been confirmed by the numerical check.
It is evident that the examples we have provided show that the use of concepts borrowed from umbral theory offer a fairly powerful tool to deal with the "harmonic based" functions.
The next step we will touch in this paper is to check whether non integer forms of harmonic numbers make any sense.
We consider indeed the following function √ h e(x) = eˆh and the integral To this aim we remind the following identity from Laplace transform theory [8] e −p the numerical check has, in both cases, confirmed the correctness of the ansatz. The possibility of defining k √ h e(x) will be discussed elsewhere.
We go back to eq. (7) and write the first derivative of the HBEF as By taking into account that h n+1 = h n + 1 n+1 we end up with the following differential equation defining the function h e(x) 3 The relevant solution reads h e(x) = 1 + e z (ln(x) + E 1 (x) + γ) , which is the generating function of harmonic numbers originally derived by Gosper (see [2]).
By iterating the previous procedure we find the following general recurrence The harmonic polynomials are easily shown to be linked to the HBEF by means of the generating function