[Paper Review] Higher Derivatives of the Falling Factorial and Related Generalizations of the Stirling and Harmonic Numbers

Steven S.S. Poon|arXiv (Cornell University)|Jan 13, 2014

Orbital Angular Momentum in Optics7 references3 citations

TL;DR

This paper derives higher-order derivatives of the falling factorial using a brute-force approach, introducing the 'elementary symmetric harmonic sum' as a natural framework for expressing these derivatives. It establishes a novel connection between these generalized harmonic numbers and r-Stirling numbers, offering a new combinatorial interpretation of higher derivatives in terms of symmetric sums and special number sequences.

ABSTRACT

Through a brute-force approach to calculating the higher derivatives of the falling factorial function, a number of interesting quantities were obtained and analyzed. In particular, it was found that a quantity that can be described as the "elementary symmetric harmonic sum" is a natural way to describe the solutions to such higher derivatives. The relationship of this type of generalized harmonic number to other quantities discussed in the literature, such as the r-Stirling numbers, was described in detail.

Motivation & Objective

• To systematically compute higher-order derivatives of the falling factorial function using direct analytical methods.
• To identify and define a new class of generalized harmonic numbers—termed 'elementary symmetric harmonic sums'—that naturally describe the structure of these derivatives.
• To establish a precise combinatorial relationship between these generalized harmonic numbers and the r-Stirling numbers, which extend classical Stirling number theory.
• To provide a unified framework for understanding higher derivatives of factorial-type functions through symmetric polynomial identities.
• To contribute new combinatorial identities and number sequences relevant to special functions and discrete mathematics.

Proposed method

• The paper employs a brute-force computation strategy to calculate the k-th derivative of the falling factorial function (x)_n = x(x-1)...(x-n+1).
• It introduces the concept of 'elementary symmetric harmonic sums' as a symmetric polynomial construction that captures the coefficients in the derivative expansion.
• The method involves expressing the k-th derivative as a linear combination of falling factorials, with coefficients derived from elementary symmetric functions of reciprocals of integers.
• The analysis connects these coefficients to known combinatorial numbers, particularly the r-Stirling numbers, through generating function and identity manipulation.
• The derivation relies on algebraic manipulation of symmetric polynomials and recursive identities to generalize classical harmonic number structures.
• The framework is validated by showing consistency with known results for lower-order derivatives and by extending them to higher orders.

Experimental results

Research questions

• RQ1 How can the higher-order derivatives of the falling factorial be systematically expressed in closed form?
• RQ2 What combinatorial structure underlies the coefficients appearing in the higher derivatives of the falling factorial?
• RQ3 In what way do the generalized harmonic numbers defined via elementary symmetric sums relate to the r-Stirling numbers?
• RQ4 Can the elementary symmetric harmonic sum serve as a unifying framework for higher derivatives and special number sequences?
• RQ5 What new identities or properties emerge when extending classical Stirling and harmonic number concepts using symmetric polynomial constructions?

Key findings

• The k-th derivative of the falling factorial (x)_n is expressible as a linear combination of lower-order falling factorials, with coefficients given by elementary symmetric harmonic sums.
• The elementary symmetric harmonic sum is defined as the k-th elementary symmetric sum of the reciprocals 1/1, 1/2, ..., 1/n, forming a natural generalization of harmonic numbers.
• This generalized harmonic number structure directly corresponds to the coefficients in the derivative expansion, providing a combinatorial interpretation of the derivative's algebraic form.
• A precise algebraic relationship is established between the generalized harmonic numbers and the r-Stirling numbers of the first kind, extending classical identities.
• The method successfully generalizes classical results on derivatives of polynomials and provides a systematic approach to higher-order differentiation of factorial functions.
• The framework enables the derivation of new combinatorial identities involving symmetric sums and factorial derivatives, enriching the theory of special number sequences.