Advances in Mathematical Physics

Volume 2018, Article ID 9575626, 9 pages

https://doi.org/10.1155/2018/9575626

## Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity

^{1}Institute of Nuclear Physics Polish Academy of Sciences, 31342 Kraków, Poland^{2}Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS UMR 7030, 93430 Villetaneuse, France^{3}Université Pierre et Marie Curie (Paris 06), Sorbonne Universités, LPTMC, CNRS UMR 7600, 75252 Paris Cedex 05, France

Correspondence should be addressed to Pawel Blasiak; lp.ude.jfi@kaisalb.lewap

Received 30 November 2017; Accepted 26 September 2018; Published 22 October 2018

Academic Editor: Andrei D. Mironov

Copyright © 2018 Pawel Blasiak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We give a purely combinatorial proof of the Glaisher-Crofton identity which is derived from the analysis of discrete structures generated by the iterated action of the second derivative. The argument illustrates the utility of symbolic and generating function methodology of modern enumerative combinatorics. The paper is meant for nonspecialists as a gentle introduction to the field of graphical calculus and its applications in computational problems.

#### 1. Introduction

Many computational problems involve action of complex expressions in derivatives on functions. A typical example is the exponential of a Hamiltonian acting on some initial condition which is a formal solution to the evolution equation. Applications of the latter range from classical heat and diffusion theory, financial mathematics, and economy to quantum field theory, hence practical interest in operational formulas enabling explicit evaluation of such expressions. Methods used to this effect usually involve operator and special function techniques, integral transforms, umbral calculus methods, etc. See comprehensive review of the subject [1, 2]. In this paper we develop another approach based on modern combinatorial methods of the analysis and enumeration of structures via generating functions [3–5].

The most known operational identities that involve the exponential of the first derivative are formulas for the shift and dilation operatorswhere is an arbitrary function. (Here, we leave subtle problems of convergence aside and consider as a formal power series in one variable .) They are a special case of the general closed-form operational expressionwhere functions and are specified by the following equationsSee [1, 2, 6] for the proof based on operator techniques and [7, Sect. 6] for a recently developed combinatorial approach.

Note that formulas (1) and (2) are valid for any function . However, this is a very unique situation which holds only for the exponential of an expression linear in the first derivative. For the second derivative the closed-form formulas are not known and the best that can be done is evaluation on specific functions. There are only a few explicit examples which include the formula for the exponential generating function of Hermite polynomials [8] and the Glaisher-Crofton identity [9–11]and the Glaisher-Crofton identity [9–11]

Formulas of this type are usually derived using integral representations in the complex domain. (For example, to derive (6) one may use the integral representation ; see [10, 11].) However, in this paper we demonstrate that it is also possible to prove these identities on the basic algebraic level by analysing combinatorial structures generated by the iterated action of the second derivative. In this approach functions are treated as generating functions enumerating simple combinatorial objects (like sets of subsets, cycles, sequences, etc.) and consequently expressions in the derivatives transform objects in the initial class into richer structures whose generating functions can be quickly identified with the methods of symbolic combinatorics [3, 4]. We remark that methodology exposed in this note treads in the steps of combinatorial approach to algebraic identities developed by D. Foata in [12, 13].

Our goal in this paper is to develop and promote general combinatorial methodology for solving computational problems which, in many cases, provides better insight into algebraic and analytic manipulations. We illustrate this approach by explaining combinatorial meaning of the exponential of the second derivative and use this interpretation to derive equations (5) and (6).

#### 2. Combinatorics of Derivatives

Action of derivatives on a function can be seen as a transformation of combinatorial structures. This viewpoint comes from interpreting the function as a generating function enumerating objects in some combinatorial class. In the following we formalise this intuition by describing the relevant constructions and develop a broader picture which includes higher derivatives and their exponentials. This framework will be illustrated by a simple combinatorial proof and interpretation of Taylor’s formula in Section 2.1 and equations (5) and (6) in Sections 2.2 and 3.

##### 2.1. Generating Functions, First Derivative, and Taylor’s Formula

Let us consider a combinatorial class which is defined as a denumerable collection of objects built of atoms represented by according to some well specified procedure. A typical combinatorial problem consists in enumeration of objects in according to the size which is usually the number of atoms. In other words, one seeks the sequence which counts the number of objects comprised of exactly atoms. This sequence can be encoded in a generating functionwhich is a convenient tool for enumeration of complex structures via the so-called transfer rules. The latter translate combinatorial constructions into algebraic manipulations of the corresponding generating functions (see [3–5] for a comprehensive treatment of the subject and the appendix for a quick extract of a few transfer rules used in this paper). In the following we will briefly review combinatorics of the first derivative and recall a simple combinatorial interpretation of Taylor’s formula.

Here we will be concerned with the derivative operation acting on some well-defined class which consists in *“selecting in all possible ways a single atom of type ** in each element of ** and replacing it with an atom of a new type **.”*

In other words, one may think of the new class as formed of all structures taken from in which one of the atoms gets “repainted” into a new colour . Since each structure in built of atoms of type gives rise to new ones with atoms and a single , then the generating function enumerating objects in the new class is given bywhere counts objects according to the number of ’s and ’s, respectively. This is substantiated in the standard transfer rule:

Now, let us consider the -th derivative acting on . Combinatorially it means *“select in all possible ways an unordered collection of ** atoms of type ** and replace (repaint) them by atoms of type **.”*

Clearly, for each structure of size in we have possible choices, and hence the generating function of the new class evaluates toIn consequence we get the following transfer rule:which gives the combinatorial interpretation of the -th derivative on the level of combinatorial structures.

This brings an interesting perspective on the derivative of a function which we will develop throughout the paper. Namely, one may think of a function as a generating function of some combinatorial class . Then differentiation yields a new generating function which enumerates objects in the new class comprised of structures taken from in which some of the atoms were replaced with ’s (how many are replaced depends on the order of the derivative). Hence, the derivative of a function can be understood as a well-defined combinatorial transformation of the associated combinatorial class in a sense that on the level of generating functions it corresponds to simple differentiation (cf. (9) and (11)).

For illustration of this viewpoint let us recall the usual Taylor’s formulaSurprisingly, it admits a transparent combinatorial interpretation (see [3, Note III.31] or [7, Note 3]). To see this we observe that the l.h.s. is the sum of derivatives applied to some function which can be considered as the generating function of some class of objects built of atoms . Then from our previous discussion the exponential corresponds to *“selecting in all possible ways an arbitrary number of **’s and replacing them by **’s*.”

(since the sum contains derivatives of arbitrary order we may choose subsets of arbitrary cardinality). On the other hand, this is the same as substituting each atom either with atom (which makes no real effect) or with atom (which means the replacement). Hence we have the following combinatorial equivalencewhich on the level of generating functions, by virtue of (11) and the transfer rule for substitutions (see the appendix, (A.4)), directly translates into (12). Hence from the combinatorial point of view Taylor’s formula is a simple manifestation of the following transfer rule (cf. (13)):which applies to any combinatorial class and its generating function . This is a typical example of combinatorial methodology which draws on the fact that in many cases the same combinatorial structure allows different specifications.

##### 2.2. Second Derivative and Hermite Polynomials

Combinatorial interpretation of the second derivative can be developed along similar lines. From the above we know that the 2-nd derivative acting on consists in *“selecting in all possible ways an unordered pair of ** atoms and replacing the chosen **’s by atoms of type **.”*

We will call such (unordered) pair a* doubleton*. Clearly, for an object composed of atoms this can be done in ways, which agrees with the algebraic identity .

More generally, by iterating times one picks out a sequence of doubletons in the original structure. Hence, we define the following construction:

which consists in *“selecting in all possible ways a set of ** unordered pairs (doubletons) of ** atoms and replacing them by ** atoms*.”

Note that, as in (11), we deem the order in the sequence irrelevant by introducing factor in front of the iterated derivative (hence the “set” and not the “sequence” in the description). For a quick check of this specification we observe that the coefficient on the r.h.s. of the identitycoincides with the number of possible ways of choosing a set of unordered pairs from the set of objects, and hence by linearity we establish correctness of the description and transfer rule (15).

Now, we are in position to give interpretation of the exponential of second derivative:whose combinatorial meaning comes down to *“selecting in all possible ways an arbitrary subset of (unordered) pairs of ** atoms and replacing each chosen ** by atom of type *.”

This is the sum of the derivative operations of the type (15) which on the level of generating functions is the sum of the derivatives. Unfortunately it does not come close to any neat expression like (12) or (14). Indeed, Taylor’s formula does not generalise in a straightforward manner. Innocuous as it may seem, selecting pairs instead of singletons introduces considerable complexity into the picture and requires careful analysis which quickly gets intractable. However, in particular cases of simple combinatorial structures (and their generating functions) it is possible to carry all the calculus through. An example that we will consider in detail is the Glaisher-Crofton formula (6) which evaluates action of the exponential of the second derivative on the gaussian. Before we proceed to this result, discussed in Section 3, we will illustrate our combinatorial methodology on a simpler case of the action on monomials and provide a link with combinatorial model of Hermite polynomials (cf. [4, 12, 13]).

Let us start with the explicit expression which is obtained from expanding the exponential and differentiating the monomial, i.e.,For it specialises to the Hermite polynomial . More generally, we may also writewhich stems from the fact that is an eigenvector of the derivative operator to eigenvalue . Again, for it is the exponential generating function of Hermite polynomials (cf. (5)). For the purpose at hand we will leave variable unspecified so as to deal with positive integers only (cf. coefficients in (18)). (It is a typical combinatorial trick to introduce additional labels (or weights) which often allows getting rid of negative or noninteger factors entering multiplicatively in the expressions. Then enumeration of structures proceeds also with respect to this additional label (or weight) which, if needed, can be specified to the required value at the end.) Our aim is to understand these formulas in terms of enumeration of structures.

In order to use combinatorial description of (17) we interpret as the exponential generating function of the labelled class of sets (see [3, Sect. II] for a precise definition and discussion of labelled classes and their relation with exponential generating functions). It is comprised of sets whose atoms carry integer labels , and additionally to each an atom (or weight) of type is attached; see Figure 1 on the left. We have the following translation rule (see (A.5)):(Clearly, for given one can built one such set and its weight is .) Now, following the combinatorial description (17) action of on an individual set in consists in selecting in all possible ways a subset of doubletons. This amounts to splitting of each original set into products of two subsets: one comprising singletons and the other doubletons. Additionally, these subsets differ in that the atoms in the singletons (untouched by the derivatives) carry the weight , while each atom forming the doubleton (arising from nontrivial action of the second derivative) carries the label . Yet another way of seeing the resulting class of objects is to understand them simply as a set of singletons and doubletons . See Figure 1 for illustration. Formally, one writes the following sequence of combinatorial equivalences:which on the level of generating functions readily transform (cf. the appendix) into a sequence of algebraic equalities providing a combinatorial proof of (19).