Umbral Methods, Combinatorial Identities and Harmonic Numbers

We analyse and demonstrate how umbral methods can be applied for the study of the problems, involving combinatorial calculus and harmonic numbers. We demonstrate their efficiency and we find the general procedure to frame new and existent identities within a unified framework, amenable of further generalizations.


Introduction
In this article we employ methods of umbral nature to provide a common framework for known and new identities regarding combinatorial calculus and harmonic numbers.
Just to give a glimpse into the technique, adopted in this article, we remind the identity [1] can be cast in the following form: 1( 1 ) 1 1 n a n   . ( Equation (1) can be written in the form (3) just as the consequence of the binomial theorem and of definition (2) and it is a useful tool to generate new identities, listed below: a) The duplication -theorem‖: The proof of this last identity is easily achieved by following the steps outlined below. We can obtain the obvious consequence of the equation ( c) The multiplication theorem: The proof of b) and c) theorems is achieved by the same procedure leading to the proof of a) and is omitted here for the sake of conciseness 1 Now let us introduce the operator of the umbral derivative, defined by the following rule: 1ˆn n a a n a   (9) which, along with the multiplication condition: 1ˆnn a a a   (10) yields the following result for the commutator bracket between the two operators: Equation (11) ensures that, we can benefit from the properties of the Weyl-Heisenberg algebra, characterising our problem. Within this framework the following simple example is provided by the definition of the associated Hermite polynomials: two variables Hermite polynomials are defined below with the variable x replacing the operator â as follows 2 : n n n r r r n r r a y y H a y n n n r r n r n r r It is easy to show that the following recurrences are satisfied: which are direct generalisations of the relevant to the ordinary Hermite polynomials relations. The umbral heat equation (14) can be exploited to define the polynomials (12) in terms of the following operational equation: In these introductory remarks we have presented few elements of the formalism, which we employ in the following chapters to further develop the method of umbral operators and obtain new identities in combinatorial calculus, involving the Euler Beta and Riemann Zeta functions.

Umbral Methods and the Euler Beta Function
Let us take note that the parameter n in the equation (1) can be treated as a variable and, therefore, p times repeated derivatives with respect to n can be taken on both sides: Now, using the following series expansion [4]: assumed to be valid also for the umbral operator â , we end up with the following identity: where   , S k m -the Stirling numbers of the first kind [4]. The validity of (18) has been checked aposteriori by a numerical procedure. Explicit study of the Stirling numbers relations with combinatorial identities can be found in [5].
Note, that identity (18)  Taking repeated derivatives of both sides of (19) with respect to λ in the point λ = 0 yields the r power of the left-hand side of equation (18), written in terms of the Beta function instead of the Stirling numbers: (25) This last result (24) represents essentially the equation (18), written without any explicit use of infinite sums. As to the explicit evaluation of the derivatives of the Beta function, we note that they possess the integral representation, provided by [6], [7]: and, therefore, we find the following expression for the derivative of the Beta function: In the next chapter we will apply the above obtained results to the theory of the harmonic numbers.

Umbral Methods and Harmonic Numbers
The harmonic numbers [8] are usually denoted by H n ; to avoid confusions with Hermite polynomials we use here h n notation: Umbral methods technique simplifies the derivation of the properties of the harmonic numbers and the study of the associated generating functions.
We can express the harmonic numbers (28) in terms of the umbral variable (2) as follows: The use of the above described procedure and of the identities derived in the previous chapter 2 yields the generalization of the formula (32): 1 ( 1 Further comments on umbral methods and harmonic numbers are given in the following concluding chapter.

Generalisations and Discussion
To complete the study of the harmonic numbers and umbral methods, consider the following sum: Employing the results of the previous chapter 3 we recast A(n) in the following operator form: differential equations.