Bell’s Polynomials and Laplace Transform of Higher Order Nested Functions

Abstract

Using Bell’s polynomials it is possible to approximate the Laplace Transform of composite functions. The same methodology can be adopted for the evaluation of the Laplace Transform of higher-order nested functions. In this case, a suitable extension of Bell’s polynomials, as previously introduced in the scientific literature, is used, namely higher order Bell’s polynomials used in the representation of the derivatives of multiple nested functions. Some worked examples are shown, and some of the polynomials used are reported in the Appendices.

1. Introduction

In this study, we illustrate a procedure for the evaluation of the Laplace Transform (LT) of multi-nested analytic functions. To this end, we make use of Bell’s polynomials [1,2,3,4,5], which constitute the essential tool for computing the subsequent derivatives of composite functions.

The Bell’s polynomials appear in many different fields, ranging from number theory [6,7,8] to operator theory [9], and from differential equations [4] to integral transforms [10,11]. It is worth noting here that Bell’s polynomials are closely related to and can be written in terms of symmetric functions in combinatorial Hopf algebras [12].

The importance of the LT [13,14] is well known and it is redundant to remind it here.

We use the classic definition of the LT:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \mathcal{L}\left(f\right):={\int }_0^{\infty }exp(-st)f\left(t\right)dt=L\left(s\right).}\end{array}\end{array}$$

The LT converts a function of a real variable t (usually representing the time) to a function of a complex variable s (which represents the complex frequency). The LT holds for locally integrable functions on $[0,+\infty )$. It is convergent in every half-plane $Re\left(s\right)>a$, where a is the so-called convergence abscissa, depending on the growth rate at infinity of $f\left(t\right)$.

Our procedure is as follows: we use Taylor’s expansion of the considered analytic function, and express the relevant coefficients in terms of Bell’s polynomials; then, we approximate the LT of the given nested function by a series expansion, which provides an asymptotic representation of the LT when that exists.

We start from the easier case of the LT of a nested exponential function, considering the first few values of the complete Bell’s polynomials. The result is a Laurent expansion which approximates the relevant LT.

Then, we consider the case of the LT of general nested functions. The main problem is to provide a table of Bell’s polynomials. These exhibit higher complexity, but their evaluation can be easily performed through a dedicated computer code.

Our results can be compared with the LT of nested functions appearing in the literature only in a few cases [15], but the results we have obtained in these cases are completely satisfactory.

All the computations reported in this study have been performed using the computer algebra program Mathematica${}^{\copyright }$.

The second-order Bell’s polynomials $Y_n^{\left[2\right]}$, representing the derivatives of nested functions of the type $f\left(g\right(h\left(t\right))$ are then introduced, and two examples of LT of these functions are given.

In Appendix A a table of the second-order Bell’s polynomials is reported.

Lastly, we give some examples to show that the same methodology can be used even for the LT of higher-order nested functions. The first few terms of the corresponding generalized Bell’s polynomials, of order 4, $Y_n^{\left[4\right]}$, are shown in Appendix B The polynomials $Y_n^{\left[7\right]}$ have been computed in the same way but are not reported here owing to the lack of space.

It is worth noting that more general extensions of Bell’s polynomials have been introduced in the past, including those appearing in the two-variable case [16], as well as the multi-variable case [17]. Since all the aforementioned extensions have been proven through the classical case, more general results could be obtained by applying the methods described in this article.

2. Definition of Bell’s Polynomials

The n-th derivative of the composite (differentiable) function $\mathsf{\Phi }\left(t\right):=f\left(g\right(t\left)\right)$, as evaluated by the chain rule, is expressed by Bell’s polynomials as follows

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_n:=D_t^n\mathsf{\Phi }\left(t\right)=Y_n(f_1,g_1;f_2,g_2;\dots ;f_n,g_n)=\sum _{k=1}^n{B}_{n,k}(g_1,g_2,\dots ,g_{n-k+1})f_k,\end{array}$$

(1)

where

$$\begin{array}{c}\hfill f_h:=D_x^h{f\left(x\right)|}_{x=g\left(t\right)},g_k:=D_t^{k}g\left(t\right).\end{array}$$

(2)

The coefficients $B_{n,k},$ for all $k=1,\dots ,n,$ are polynomials of the variables $g_1,g_2,\dots ,$$g_{n-k+1},$ that are homogeneous of degree k and isobaric of weight n (i.e., they are a linear combination of monomials $g_1^{k_1}{g}_2^{k_2}\dots g_n^{k_n}$ whose weight is constantly given by $k_1+2k_2+\dots +n{k}_n=n$); in the literature, they are also referred to as partial Bell’s polynomials.

Bell’s polynomials satisfy the recursion

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{Y}_0:=f_1;\hfill \\ Y_{n+1}(f_1,g_1;\dots ;f_n,g_n;f_{n+1},g_{n+1})=\hfill \\ {\displaystyle =\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)Y_{n-k}(f_2,g_1;f_3,g_2;\dots ;f_{n-k+1},g_{n-k})g_{k+1}.}\hfill \end{array}\right.\end{array}$$

(3)

An explicit representation is given by the Faà di Bruno’s formula

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle Y_n(f_1,g_1;f_2,g_2;\dots ;f_n,g_n)=\sum _{\pi \left(n\right)}\frac{n!}{r_1!r_2!\dots r_n!}{f}_r{\left[\frac{g_1}{1!}\right]}^{r_1}{\left[\frac{g_2}{2!}\right]}^{r_2}\dots {\left[\frac{g_n}{n!}\right]}^{r_n},}\end{array}\end{array}$$

(4)

where the sum runs over all the partitions $\pi \left(n\right)$ of the integer n, $r_i$ denotes the number of parts of size i, and $r=r_1+r_2+\dots +r_n$ denotes the number of parts of the considered partition [5].

The $B_{n,k}$ coefficients satisfy the recursion $\forall n$

$$\begin{array}{c}\hfill \begin{array}{c}{B}_{n,1}=g_n,B_{n,n}=g_1^n,\\ {\displaystyle B_{n,k}(g_1,g_2,\dots ,g_{n-k+1})=\sum _{h=0}^{n-k}\left(\genfrac{}{}{0pt}{}{n-1}{h}\right)B_{n-h-1,k-1}(g_1,g_2,\dots ,g_{n-k-h+1})g_{h+1}.}\end{array}\end{array}$$

(5)

3. LT of Composite Functions

Let $f\left(g\right(t\left)\right)$ be a composite function that is analytic in a neighborhood of the origin, and whose Taylor’s expansion is given by

$$\begin{array}{c}\hfill {\displaystyle f\left(g\left(t\right)\right)=\sum _{n=0}^{\infty }{a}_n\frac{t^n}{n!},a_n=D_t^n{\left[f\left(g\left(t\right)\right)\right]}_{t=0}.}\end{array}$$

(6)

According to the preceding equations, it results in

$$\begin{array}{c}\begin{array}{c}{a}_0=f\left({\stackrel{\circ }{g}}_0\right),\hfill \\ {\displaystyle a_n=D_t^n{\left[f\left(g\left(t\right)\right)\right]}_{t=0}=\sum _{k=1}^n{B}_{n,k}({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_{n-k+1}){\stackrel{\circ }{f}}_k,(n\ge 1),}\hfill \end{array}\hfill \end{array}$$

(7)

where

$$\begin{array}{c}\hfill {\stackrel{\circ }{f}}_k:=D_x^{k}f\left(x\right){{|}_{x=g\left(0\right)},{\stackrel{\circ }{g}}_h:=D_t^{h}g\left(t\right)|}_{t=0}.\end{array}$$

(8)

Then, the following result easily follows.

Theorem 1.

Consider a composite function $f\left(g\right(t\left)\right)$ that is analytic in a neighborhood of the origin, and can be expressed by Taylor’s expansion in (6). For its LT the following asymptotic representation holds

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{+\infty }f\left(g\left(t\right)\right)e^{-ts}dt\simeq \frac{f({\stackrel{\circ }{g}}_0)}{s}+\sum _{n=1}^N{\int }_0^{+\infty }\sum _{k=1}^n{B}_{n,k}({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_{n-k+1}){\stackrel{\circ }{f}}_k\frac{t^n}{n!}{e}^{-ts}dt=}\\ {\displaystyle =\frac{f({\stackrel{\circ }{g}}_0)}{s}+\sum _{n=1}^N\left({\displaystyle \sum _{k=1}^n{B}_{n,k}({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_{n-k+1}){\stackrel{\circ }{f}}_k}\right){\int }_0^{+\infty }\frac{t^n}{n!}{e}^{-ts}dt=}\\ {\displaystyle =\frac{f({\stackrel{\circ }{g}}_0)}{s}+\sum _{n=1}^N\left({\displaystyle \sum _{k=1}^n{B}_{n,k}({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_{n-k+1}){\stackrel{\circ }{f}}_k}\right)\frac{1}{s^{n+1}},}\end{array}\end{array}$$

(9)

where N denotes a finite expansion order.

3.1. The Particular Case of the Exponential Function

In the particular case when $f\left(x\right)=e^x$, that is considering the function $e^{g\left(t\right)}$, and assuming $g\left(0\right)=0$, we have the simple form

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \sum _{k=1}^n{B}_{n,k}({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_{n-k+1}){\stackrel{\circ }{f}}_k=\sum _{k=1}^n{B}_{n,k}({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_{n-k+1})=B_n({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_n),}\end{array}\end{array}$$

(10)

where the $B_n$ are the complete Bell’s polynomials. It results $B_0\left(g_0\right):=f\left(g_0\right)$, and the first few values of $B_n$, for $n=1,2,\dots ,5$, are given by

$$\begin{array}{c}{B}_1=g_1,\hfill \\ \\ B_2=g_1^2+g_2,\hfill \\ \\ B_3=g_1^3+3g_1{g}_2+g_3,\hfill \\ \\ B_4=g_1^4+6g_1^2{g}_2+4g_1{g}_3+3g_2^2+g_4,\hfill \\ \\ B_5=g_1^5+10g_1^3{g}_2+15g_1{g}_2^2+10g_1^2{g}_3+10g_2{g}_3+5g_1{g}_4+g_5,\hfill \\ \\ B_6=g_1^6+15g_1^4{g}_2+45g_1^2{g}_2^2+15g_2^3+20g_1^3{g}_3+60g_1{g}_2{g}_3+10g_3^2+15g_1^2{g}_4+15g_2{g}_4+6g_1{g}_5+g_6,\hfill \\ \\ B_7=g_1^7+21g_1^5{g}_2+105g_1^3{g}_2^2+105g_1{g}_2^3+35g_1^4{g}_3+210g_1^2{g}_2{g}_3+105g_2^2{g}_3+70g_1{g}_3^2+35g_1^3{g}_4+\hfill \\ 105g_1{g}_2{g}_4+35g_3{g}_4+21g_1^2{g}_5+21g_2{g}_5+7g_1{g}_6+g_7,\hfill \\ \\ B_8=g_1^8+28g_1^6{g}_2+210g_1^4{g}_2^2+420g_1^2{g}_2^3+105g_2^4+56g_1^5{g}_3+560g_1^3{g}_2{g}_3+840g_1{g}_2^2{g}_3+280g_1^2{g}_3^2+\hfill \\ 280g_2{g}_3^2+70g_1^4{g}_4+420g_1^2{g}_2{g}_4+210g_2^2{g}_4+280g_1{g}_3{g}_4+35g_4^2+56g_1^3{g}_5+168g_1{g}_2{g}_5+\hfill \\ 56g_3{g}_5+28g_1^2{g}_6+28g_2{g}_6+8g_1{g}_7+g_8,\hfill \\ \\ B_9=g_1^9+36g_1^7{g}_2+378g_1^5{g}_2^2+1260g_1^3{g}_2^3+945g_1{g}_2^4+84g_1^6{g}_3+1260g_1^4{g}_2{g}_3+3780g_1^2{g}_2^2{g}_3+\hfill \\ 1260g_2^3{g}_3+840g_1^3{g}_3^2+2520g_1{g}_2{g}_3^2+280g_3^3+126g_1^5{g}_4+1260g_1^3{g}_2{g}_4+1890g_1{g}_2^2{g}_4+\hfill \\ 1260g_1^2{g}_3{g}_4+1260g_2{g}_3{g}_4+315g_1{g}_4^2+126g_1^4{g}_5+756g_1^2{g}_2{g}_5+378g_2^2{g}_5+504g_1{g}_3{g}_5+\hfill \\ 126g_4{g}_5+84g_1^3{g}_6+252g_1{g}_2{g}_6+84g_3{g}_6+36g_1^2{g}_7+36g_2{g}_7+9g_1{g}_8+g_9,\hfill \\ \\ B_{10}=g_1^{10}+45g_1^8{g}_2+630g_1^6{g}_2^2+3150g_1^4{g}_2^3+4725g_1^2{g}_2^4+945g_2^5+120g_1^7{g}_3+2520g_1^5{g}_2{g}_3+\hfill \\ 12600g_1^3{g}_2^2{g}_3+12600g_1{g}_2^3{g}_3+2100g_1^4{g}_3^2+12600g_1^2{g}_2{g}_3^2+6300g_2^2{g}_3^2+2800g_1{g}_3^3+\hfill \\ 210g_1^6{g}_4+3150g_1^4{g}_2{g}_4+9450g_1^2{g}_2^2{g}_4+3150g_2^3{g}_4+4200g_1^3{g}_3{g}_4+12600g_1{g}_2{g}_3{g}_4+\hfill \\ 2100g_3^2{g}_4+1575g_1^2{g}_4^2+1575g_2{g}_4^2+252g_1^5{g}_5+2520g_1^3{g}_2{g}_5+3780g_1{g}_2^2{g}_5+2520g_1^2{g}_3{g}_5+\hfill \\ 2520g_2{g}_3{g}_5+1260g_1{g}_4{g}_5+126g_5^2+210g_1^4{g}_6+1260g_1^2{g}_2{g}_6+630g_2^2{g}_6+840g_1{g}_3{g}_6+\hfill \\ 210g_4{g}_6+120g_1^3{g}_7+360g_1{g}_2{g}_7+120g_3{g}_7+45g_1^2{g}_8+45g_2{g}_8+10g_1{g}_9+g_{10}.\hfill \end{array}$$

The values of the complete Bell’s polynomials for particular choices of the relevant parameters can be found in [6].

The complete Bell’s polynomials satisfy the identity (see, e.g., [4])

$$\begin{array}{c}\hfill {\displaystyle B_{n+1}(g_1,\dots ,g_{n+1})=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_{n-k}(g_1,\dots ,g_{n-k})g_{k+1}.}\end{array}$$

(11)

In this case Equation (9) reduces to

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{+\infty }exp\left(g\left(t\right)\right)e^{-ts}dt\simeq \frac{exp\left({\stackrel{\circ }{g}}_0\right)}{s}+\sum _{n=1}^N{B}_n({\stackrel{\circ }{g}}_1,{\stackrel{\circ }{g}}_2,\dots ,{\stackrel{\circ }{g}}_n)\frac{1}{s^{n+1}}.}\end{array}\end{array}$$

(12)

In what follows, we evaluate the approximation of the LT of nested functions. The reported results have been obtained using the computer algebra program Mathematica${}^{\copyright }$.

Examples

We first recall the case of the LT of nested exponential functions, showing two particular examples.

• Consider the Bessel function $g\left(t\right):=J_1\left(t\right)$ and the LT of the corresponding exponential function. We find

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{+\infty }exp\left(J_1\left(t\right)\right)e^{-ts}dt=\frac{1}{s}+\frac{1}{2s^2}+\frac{1}{4s^3}-\frac{3}{4s^4}-\frac{11}{16s^5}-\frac{19}{32s^6}+\frac{91}{64s^7}+}\\ {\displaystyle +\frac{701}{128s^8}+\frac{953}{256s^9}-\frac{15245}{512s^{10}}+O\left({\displaystyle \frac{1}{s^{11}}}\right),}\end{array}\end{array}$$

(13)

for $s\to \infty$.

• Consider the function $g\left(t\right):=arctan\left(t\right)$ and the LT of the corresponding exponential function. We find

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{+\infty }exp(arctan\left(t\right))e^{-ts}dt=\frac{1}{s}-\frac{1}{s^2}+\frac{1}{s^3}+\frac{1}{s^4}-\frac{7}{s^5}-\frac{5}{s^6}+\frac{145}{s^7}+}\\ {\displaystyle +\frac{5}{s^8}-\frac{6095}{s^9}+\frac{5815}{s^{10}}+O\left({\displaystyle \frac{1}{s^{11}}}\right),}\end{array}\end{array}$$

(14)

for $s\to \infty$.

4. LT in Two Known Cases

We considered two cases concerning composite functions whose transform and anti-transform are known (see [15]). By using the computer algebra program Mathematica${}^{\copyright }$, we have been able to prove the correctness of the methodology used.

4.1. Case #1

Consider the function $l\left(t\right)=log[cosh(t\left)\right]$. The LT of $l\left(t\right)$ is found to be [15]:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle L\left(s\right)=\frac{1}{2s}\left[\psi \left(\frac{1}{2}+\frac{s}{4}\right)-\psi \left(\frac{s}{4}\right)\right]-\frac{1}{s^2},}\end{array}\end{array}$$

(15)

for $\Re s>0$, and where $\psi \left(z\right)$ is the logarithmic derivative of the gamma function, given by

$$\begin{array}{c}\hfill \begin{array}{c}\psi \left(z\right)\equiv \frac{d}{dz}ln\mathsf{\Gamma }\left(z\right)=\frac{{\mathsf{\Gamma }}^{\prime }\left(z\right)}{\mathsf{\Gamma }\left(z\right)}.\end{array}\end{array}$$

(16)

Using our methodology, we find that

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle L\left(s\right)\simeq \tilde{L}\left(s\right)=\frac{1}{s^3}-\frac{2}{s^5}+\frac{16}{s^7}-\frac{272}{s^9}+\frac{7936}{s^{11}},}\end{array}\end{array}$$

(17)

so that, the inverse Laplace transformation is given by

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{l}\left(t\right)\simeq \left({\displaystyle \frac{t^2}{2}-\frac{t^4}{12}+\frac{t^6}{45}-\frac{17t^8}{2520}+\frac{31t^{10}}{14175}}\right)H\left(t\right),\end{array}\end{array}$$

(18)

with $H\left(·\right)$ denoting the Heaviside distribution which can be defined as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle H\left(x\right)={\int }_{-\infty }^x\delta \left(u\right)du,}\end{array}\end{array}$$

(19)

in terms of the Dirac delta distribution $\delta \left(·\right)$.

4.2. Case #2

Let us consider the function $l\left(t\right)=J_0\left(t^2\right)$. The LT of $l\left(t\right)$ is found to be [15]:

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle L\left(s\right)=\frac{\pi s}{16}\left\{{\left[J_{1/4}(s^2/8)\right]}^2+{\left[Y_{1/4}(s^2/8)\right]}^2\right\},}\end{array}\end{array}$$

(20)

for $\Re s>0$.

Using our methodology, we find that

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle L\left(s\right)\simeq \tilde{L}\left(s\right)=\frac{1}{s}-\frac{6}{s^5}+\frac{630}{s^9}-\frac{207900}{s^{13}}+\frac{141891750}{s^{17}}-\frac{164991726900}{s^{21}},}\end{array}\end{array}$$

(21)

so that, the inverse Laplace transformation is given by

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{l}\left(t\right)\simeq \left({\displaystyle 1-\frac{t^4}{4}+\frac{t^8}{64}-\frac{t^{12}}{2304}+\frac{t^{16}}{147456}-\frac{t^{20}}{14745600}}\right)H\left(t\right).\end{array}\end{array}$$

(22)

5. A First Extension of Bell’s Polynomials

We consider the second-order Bell’s polynomials, $Y_n^{\left[2\right]}(f_1,g_1,h_1;f_2,g_2,h_2;\dots ;f_n,g_n,h_n)$, defined by the n-th derivative of the composite function $\mathsf{\Phi }\left(t\right):=f\left(g\right(h\left(t\right)\left)\right)$.

Consider the functions $x=h\left(t\right)$, $z=g\left(x\right)$, and $y=f\left(z\right)$, and suppose that $h\left(t\right)$, $g\left(x\right)$, and $f\left(z\right)$ are n times differentiable with respect to their variables, so that the composite function $\mathsf{\Phi }\left(t\right):=f\left(g\right(h\left(t\right)\left)\right)$ can be differentiated n times with respect to t, by using the chain rule.

We use, as before, the following notation:

$${\mathsf{\Phi }}_j:=D_t^j\mathsf{\Phi }\left(t\right),f_h:=D_y^{h}f\left(y\right){{|}_{y=g\left(x\right)},g_k:=D_x^{k}g\left(x\right)|}_{x=h\left(t\right)},h_r:=D_t^{r}h\left(t\right).$$

Then, the n-th derivative can be represented by the compact symbol:

$$\begin{array}{c}\hfill \begin{array}{c}{\mathsf{\Phi }}_n=Y_n^{\left[2\right]}(f_1,g_1,h_1;f_2,g_2,h_2;\dots ;f_n,g_n,h_n)=Y_n^{\left[2\right]}\left({[f,g,h]}_n\right),\end{array}\end{array}$$

(23)

where the $Y_n^{\left[2\right]}$ are defined as the second order Bell’s polynomials.

The first few terms are as follows.

$$\begin{array}{c}\hfill \begin{array}{c}{Y}_1^{\left[2\right]}\left({[f,g,h]}_1\right)=f_1{g}_1{h}_1;\hfill \\ Y_2^{\left[2\right]}\left({[f,g,h]}_2\right)=f_1{g}_1{h}_2+f_1{g}_2{h}_1^2+f_2{g}_1^2{h}_1^2;\hfill \\ Y_3^{\left[2\right]}\left({[f,g,h]}_3\right)=f_1{g}_1{h}_3+f_1{g}_3{h}_1^3+3f_1{g}_2{h}_1{h}_2+3f_2{g}_1{g}_2{h}_1^3+f_3{g}_1^3{h}_1^3;\hfill \\ Y_4^{\left[2\right]}\left({[f,g,h]}_4\right)=f_4{g}_1^4{h}_1^4+6f_3{g}_1^2{g}_2{h}_1^4+3f_2{g}_2^2{h}_1^4+4f_2{g}_1{g}_3{h}_1^4+f_1{g}_4{h}_1^4+6f_3{g}_1^3{h}_1^2{h}_2+\hfill \\ +18f_2{g}_1{g}_2{h}_1^2{h}_2+6f_1{g}_3{h}_1^2{h}_2+3f_2{g}_1^2{h}_2^2+3f_1{g}_2{h}_2^2+4f_2{g}_1^2{h}_1{h}_3+4f_1{g}_2{h}_1{h}_3+f_1{g}_1{h}_4;\hfill \\ Y_5^{\left[2\right]}\left({[f,g,h]}_5\right)=f_5{g}_1^5{h}_1^5+10f_4{g}_1^3{g}_2{h}_1^5+15f_3{g}_1{g}_2^2{h}_1^5+10f_3{g}_1^2{g}_3{h}_1^5+10f_2{g}_2{g}_3{h}_1^5+\hfill \\ +5f_2{g}_1{g}_4{h}_1^5+f_1{g}_5{h}_1^5+10f_4{g}_1^4{h}_1^3{h}_2+60f_3{g}_1^2{g}_2{h}_1^3{h}_2+30f_2{g}_2^2{h}_1^3{h}_2+40f_2{g}_1{g}_3{h}_1^3{h}_2+\hfill \\ +10f_1{g}_4{h}_1^3{h}_2+15f_3{g}_1^3{h}_1{h}_2^2+45f_2{g}_1{g}_2{h}_1{h}_2^2+15f_1{g}_3{h}_1{h}_2^2+10f_3{g}_1^3{h}_1^2{h}_3+30f_2{g}_1{g}_2{h}_1^2{h}_3+\hfill \\ +10f_1{g}_3{h}_1^2{h}_3+10f_2{g}_1^2{h}_2{h}_3+10f_1{g}_2{h}_2{h}_3+5f_2{g}_1^2{h}_1{h}_4+5f_1{g}_2{h}_1{h}_4+f_1{g}_1{h}_5.\hfill \end{array}\end{array}$$

A more extended table is given in Appendix A.

The connections to the ordinary Bell’s polynomials are highlighted below.

Theorem 2.

For every integer n, the polynomials $Y_n^{\left[2\right]}$ are represented in terms of the ordinary Bell’s polynomials by the following equation, where a compact notation similar to the one in (23) is used:

$$\begin{array}{c}\hfill \begin{array}{c}{Y}_n^{\left[2\right]}\left({[f,g,h]}_n\right)=\hfill \\ =Y_n\left(f_1,Y_1\left({[g,h]}_1\right);f_2,Y_2\left({[g,h]}_2\right);\dots ;f_n,Y_n\left({[g,h]}_n\right)\right)\hfill \end{array}\end{array}$$

(24)

Proof. 

Using induction, we can conclude that (24) is true for $n=1$, since

$$Y_1^{\left[2\right]}\left({[f,g,h]}_1\right)=f_1{g}_1{h}_1=f_1{Y}_1\left({[g,h]}_1\right)=Y_1\left(f_1,Y_1\left({[g,h]}_1\right)\right).$$

Then, assuming that (24) is true for every n, it follows that

$$\begin{array}{c}\hfill \begin{array}{c}{Y}_{n+1}^{\left[2\right]}\left({[f,g,h]}_{n+1}\right)=D_t{Y}_n^{\left[2\right]}\left({[f,g,h]}_n\right)=D_t{Y}_n\left(f_1,Y_1\left({[g,h]}_1\right);\dots ;f_n,Y_n\left({[g,h]}_n\right)\right)=\\ =Y_{n+1}\left(f_1,Y_1\left({[g,h]}_1\right);f_2,Y_2\left({[g,h]}_2\right);\dots ;f_{n+1},Y_{n+1}\left({[g,h]}_{n+1}\right)\right).\end{array}\end{array}$$

(25)

□

Consequently, we have the theorem:

Theorem 3.

The second-order Bell’s polynomials verify the recursion

$$\begin{array}{c}\hfill \begin{array}{c}{Y}_0^{\left[2\right]}=f_1;\hfill \\ Y_{n+1}^{\left[2\right]}\left({[f,g,h]}_{n+1}\right)=\hfill \\ {\displaystyle =\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)Y_{n-k}^{\left[2\right]}\left(f_2,g_1,h_1;f_3,g_2,h_2;\dots ;f_{n-k+1},g_{n-k},h_{n-k}\right)Y_{k+1}\left({[g,h]}_{k+1}\right).}\hfill \end{array}\end{array}$$

(26)

Proof. 

By means of (24) we express $Y_{n+1}^{\left[2\right]}\left({[f,g,h]}_{n+1}\right)$ in terms of

$$Y_{n+1}\left(f_1,Y_1\left({[g,h]}_1\right);\dots ;f_{n+1},Y_{n+1}\left({[g,h]}_{n+1}\right)\right).$$

Then, by using the recursion (9) and again Equation (24), the expression (26) follows. □

6. LT of Second-Order Nested Functions

Let be $f\left(g\right(h\left(t\right)\left)\right)$ be a composite function that is analytic in a neighborhood of the origin and, therefore, can be expressed by the Taylor’s expansion

$$\begin{array}{c}\hfill {\displaystyle f(g\left(\left(h\left(t\right)\right)\right)=\sum _{n=0}^{\infty }{a}_n\frac{t^n}{n!},a_n=D_t^n{[f\left(g\left(\left(h\left(t\right)\right)\right)\right]}_{t=0}.}\end{array}$$

(27)

According to the preceding equations, it results

$$\begin{array}{c}\hfill \begin{array}{c}{a}_0={\stackrel{\circ }{f}}_0=f(g\left(h\left(0\right)\right),\hfill \\ a_n=D_t^n{[f\left(g\left(\left(h\left(t\right)\right)\right)\right]}_{t=0}=Y_n^{\left[2\right]}({[\stackrel{\circ }{f},\stackrel{\circ }{g},\stackrel{\circ }{h}]}_n),(n\ge 1),\hfill \end{array}\end{array}$$

(28)

where

$$\begin{array}{c}\hfill {\stackrel{\circ }{f}}_h:=D_x^h{f\left(y\right)|}_{y=g\left(0\right)},{\stackrel{\circ }{g}}_k:=D_t^{k}g\left(x\right){{|}_{x=h\left(0\right)},{\stackrel{\circ }{h}}_r:=D_t^{r}h\left(t\right)|}_{t=0}.\end{array}$$

(29)

This expansion can be used to evaluate the LT of analytic nested functions.

Theorem 4.

Consider a nested function $f\left(g\right(\left(h\right(t\left)\right))$ that is analytic in a neighborhood of the origin, and whose Taylor’s expansion is given by (27). For its LT, the following asymptotic representation holds

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{+\infty }f(g\left(\left(h\left(t\right)\right)\right)e^{-ts}dt\simeq \frac{{\stackrel{\circ }{f}}_0}{s}+\sum _{n=1}^N{Y}_n^{\left[2\right]}({[\stackrel{\circ }{f},\stackrel{\circ }{g},\stackrel{\circ }{h}]}_n){\int }_0^{+\infty }\frac{t^n}{n!}{e}^{-ts}dt=}\\ {\displaystyle =\frac{{\stackrel{\circ }{f}}_0}{s}+\sum _{n=1}^N{Y}_n^{\left[2\right]}({[\stackrel{\circ }{f},\stackrel{\circ }{g},\stackrel{\circ }{h}]}_n)\frac{1}{s^{n+1}},}\end{array}\end{array}$$

(30)

where N denotes a finite expansion order.

Proof. 

It is a straightforward application of the definition of the second-order Bell’s polynomials. □

Example 1.

• Assuming $f\left(x\right)=e^{x-1}$, $g\left(y\right)=cos\left(y\right)$, $h\left(t\right)=sin\left(t\right)$, it results in (see Figure 1)

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{+\infty }exp[cos(sin\left(t\right))-1]e^{-ts}dt=\frac{1}{s}-\frac{1}{s^3}+\frac{8}{s^5}-\frac{127}{s^7}+\frac{3523}{s^9}-\frac{146964}{s^{11}}+O\left({\displaystyle \frac{1}{s^{13}}}\right),}\end{array}\end{array}$$

(31)

for $s\to \infty$. The corresponding inverse LT is approximated by (see Figure 2)

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{l}\left(t\right)\simeq \left({\displaystyle 1-\frac{1}{2}{t}^2+\frac{1}{3}{t}^4-\frac{127}{720}{t}^6+\frac{3523}{40320}{t}^8-\frac{12247}{302400}{t}^{10}}\right)H\left(t\right).\end{array}\end{array}$$

(32)

Figure 1. Magnitude (a) and argument (b) of the Laplace transform of $exp[cos(sin\left(t\right))-1]$ as evaluated through the approximant $\tilde{L}\left(s\right)$ and the rigorous integral expression $L\left(s\right)$ for $s=5+\mathrm{i}\omega$.

Figure 2. Distribution of $l\left(t\right)=exp[cos(sin\left(t\right))-1]$ and the relevant approximant $\tilde{l}\left(t\right)$.

Example 2.

• Upon assuming $f\left(x\right)=log\left(1+\frac{x}{2}\right)$, $g\left(y\right)=cosh\left(y\right)-1$, $h\left(t\right)=sin\left(t\right)$, it results in (see Figure 3)

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{+\infty }log\left[1+\frac{cosh(sin(t\left)\right)-1}{2}\right]e^{-ts}dt=\frac{1}{2s^3}-\frac{9}{4s^5}-\frac{27}{2s^7}+\frac{1169}{8s^9}-\frac{5869}{2s^{11}}+O\left({\displaystyle \frac{1}{s^{13}}}\right),}\end{array}\end{array}$$

(33)

for $s\to \infty$. The corresponding inverse LT can be approximated as (see Figure 4)

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{l}\left(t\right)\simeq \left({\displaystyle \frac{1}{4}{t}^2-\frac{3}{32}{t}^4+\frac{3}{160}{t}^6-\frac{167}{46080}{t}^8+\frac{5869}{7257600}{t}^{10}}\right)H\left(t\right).\end{array}\end{array}$$

(34)

Figure 3. Magnitude (a) and argument (b) of the Laplace transform of $log\left[1+\frac{cosh(sin(t\left)\right)-1}{2}\right]$ as evaluated through the approximant $\tilde{L}\left(s\right)$ and the rigorous integral expression $L\left(s\right)$ for $s=5+\mathrm{i}\omega$.

Figure 4. Distribution of $l\left(t\right)=log\left[1+\frac{cosh(sin(t\left)\right)-1}{2}\right]$ and the relevant approximant $\tilde{l}\left(t\right)$.

7. Higher Order Bell’s Polynomials

Consider the nested function $\mathsf{\Phi }\left(t\right):=f_{\left(1\right)}\left(f_{\left(2\right)}(\cdots \left(f_{\left(M\right)}\left(t\right)\right))\right)$, i.e., the composition of the functions $x_{M-1}=f_{\left(M\right)}\left(t\right)$, …, $x_1=f_{\left(2\right)}\left(x_2\right)$, $y=f_{\left(1\right)}\left(x_1\right)$, and suppose that $f_{\left(M\right)}$, …, $f_{\left(2\right)}$, $f_{\left(1\right)}$ are n times differentiable with respect to their independent variables. Then, $\mathsf{\Phi }\left(t\right)$ can be differentiated n times with respect to t using the chain rule. By definition we put $x_M:=t$, so that $y=\mathsf{\Phi }\left(t\right)$.

We use the following notation:

$$\begin{array}{c}\hfill \begin{array}{c}{\mathsf{\Phi }}_h:=D_t^h\mathsf{\Phi }\left(t\right),\hfill \\ f_{\left(1\right),h}:=D_{x_1}^h{f}_{\left(1\right)}{|}_{x_1=f_{\left(2\right)}(f_{\left(3\right)}(\dots (f_{\left(M\right)}\left(t\right))))},\hfill \\ f_{\left(2\right),k}:=D_{x_2}^k{f}_{\left(2\right)}{|}_{x_2=f_{\left(3\right)}(f_{\left(4\right)}(\dots (f_{\left(M\right)}\left(t\right))))},\hfill \\ ..............\hfill \\ f_{\left(M\right),j}:=D_{x_M}^j{f}_{\left(M\right)}{|}_{x_M=t}.\hfill \end{array}\end{array}$$

(35)

Then, the n-th derivative can be represented as

$${\mathsf{\Phi }}_n=Y_n^{[M-1]}(f_{\left(1\right),1},\dots ,f_{\left(M\right),1};f_{\left(1\right),2},\dots ,f_{\left(M\right),2};\dots ;f_{\left(1\right),n},\dots ,f_{\left(M\right),n}),$$

where the $Y_n^{[M-1]}$ are, by definition, Bell’s polynomials of order $M-1$.

The above Theorems 2 and 3 can be generalized as follows.

Theorem 5.

For every integer n, the polynomials $Y_n^{[M-1]}$ are expressed in terms of Bell’s polynomials of a lower order, through the following equation:

$$\begin{array}{c}\hfill \begin{array}{c}{Y}_n^{[M-1]}(f_{\left(1\right),1},\dots ,f_{\left(M\right),1};\dots ;f_{\left(1\right),n},\dots ,f_{\left(M\right),n})=\hfill \\ =Y_n(f_{\left(1\right),1},Y_1^{[M-2]}(f_{\left(2\right),1},\dots ,f_{\left(M\right),1});\hfill \\ f_{\left(1\right),2},Y_2^{[M-2]}(f_{\left(2\right),1},\dots ,f_{\left(M\right),1};f_{\left(2\right),2},\dots ,f_{\left(M\right),2});\dots \hfill \\ \left.\dots ;f_{\left(1\right),n},Y_n^{[M-2]}(f_{\left(2\right),1},\dots ,f_{\left(M\right),1};\dots ;f_{\left(2\right),n},\dots ,f_{\left(M\right),n})\right).\hfill \end{array}\end{array}$$

(36)

Theorem 6.

The following recurrence relation for the Bell’s polynomials $Y_n^{[M-1]}$ of order $M-1$ holds true:

$$\begin{array}{c}\hfill \begin{array}{c}{Y}_0^{[M-1]}=f_{\left(1\right),1};\hfill \\ Y_{n+1}^{[M-1]}(f_{\left(1\right),1},\dots ,f_{\left(M\right),1};\dots ;f_{\left(1\right),n+1},\dots ,f_{\left(M\right),n+1})=\hfill \\ {\displaystyle =\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)Y_{n-k}^{[M-1]}\left(f_{\left(1\right),2},f_{\left(2\right),1},\dots ,f_{\left(M\right),1};f_{\left(1\right),3},f_{\left(2\right),2},\dots ,f_{\left(M\right),2};\right.\dots }\hfill \\ \left.\dots ;f_{\left(1\right),n-k+1},f_{\left(2\right),n-k},\dots ,f_{\left(M\right),n-k}\right)\times \hfill \\ \times Y_{k+1}^{[M-2]}\left(f_{\left(2\right),1},\dots ,f_{\left(M\right),1};\dots ;f_{\left(2\right),k+1},\dots ,f_{\left(M\right),k+1}\right).\hfill \end{array}\end{array}$$

(37)

Example 3.

We apply the above results to the case of the LT of nested sine functions, assuming $M=4$ and $M=7$.

• Let be $M=4$. We have (see Figure 5):

$$f_4\left(t\right)=f_3\left(t\right)=f_2\left(t\right)=f_1\left(t\right)=sin\left(t\right),f\left(t\right)=sin(sin(sin(sin\left(t\right)))),$$

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{\infty }exp(-st)sin(sin(sin(sin\left(t\right))))dt=\frac{1}{s^2}-\frac{4}{s^4}+\frac{64}{s^6}-\frac{2160}{s^8}+\frac{121600}{s^{10}}+O\left({\displaystyle \frac{1}{s^{12}}}\right),}\end{array}\end{array}$$

Figure 5. Magnitude (a) and argument (b) of the Laplace transform of $sin(sin(sin(sin(t\left)\right)\left)\right)$ as evaluated through the approximant $\tilde{L}\left(s\right)$ and the rigorous integral expression $L\left(s\right)$ for $s=10+\mathrm{i}\omega$.

for $s\to \infty$.

The corresponding inverse LT is approximated by (see Figure 6).

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{l}\left(t\right)\simeq \left({\displaystyle t-\frac{2}{3}{t}^3+\frac{8}{15}{t}^5-\frac{3}{7}{t}^7+\frac{190}{567}{t}^9}\right)H\left(t\right).\end{array}\end{array}$$

(38)

Figure 6. Distribution of $l\left(t\right)=sin(sin(sin(sin(t\left)\right)\left)\right)$ and the relevant approximant $\tilde{l}\left(t\right)$.

• Let be $M=7$. We have (see Figure 7):

$$f_7\left(t\right)=f_6\left(t\right)=\cdots =f_1\left(t\right)=sin\left(t\right),f\left(t\right)=sin(sin(\dots sin(sin\left(t\right)))),$$

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{\infty }exp(-st)f\left(t\right)dt=\frac{1}{s^2}-\frac{7}{s^4}+\frac{217}{s^6}-\frac{14903}{s^8}+\frac{1776817}{s^{10}}+O\left({\displaystyle \frac{1}{s^{12}}}\right),}\end{array}\end{array}$$

Figure 7. Magnitude (a) and argument (b) of the Laplace transform of $sin(sin(\dots (sin(t\left)\right)\left)\right)$ as evaluated through the approximant $\tilde{L}\left(s\right)$ and the rigorous integral expression $L\left(s\right)$ for $s=10+\mathrm{i}\omega$.

for $s\to \infty$.

The corresponding inverse LT is approximated by (see Figure 8).

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{l}\left(t\right)\simeq \left({\displaystyle t-\frac{7}{6}{t}^3+\frac{217}{120}{t}^5-\frac{2129}{720}{t}^7+\frac{253831}{51840}{t}^9}\right)H\left(t\right).\end{array}\end{array}$$

(39)

Figure 8. Distribution of $l\left(t\right)=sin(sin(\dots (sin(t\left)\right)\left)\right)$ and the relevant approximant $\tilde{l}\left(t\right)$.

8. Conclusions

We have presented a method for approximating the integral of analytic composite functions. We started from the Taylor expansion of the considered function in a neighborhood of the origin. Since the coefficients can be expressed in terms of Bell’s polynomials, the integral is reduced to the computation of an approximating series, which obviously converges if the integral is convergent. Then, this methodology has been applied to the case of the LT of an analytic composite function, starting from the case of analytic nested exponential functions. Furthermore, the evaluation of the LT of analytic nested functions is discussed, and the first few second-order Bell’s polynomials used in the framework of the presented methodology are reported in Appendix A, whereas those of order 4 are given in Appendix B. A graphical verification of the proposed technique, performed in the case when the analytical forms of both the transform and anti-transform are known, proved the correctness of our results. In future studies, attention will be devoted to the evaluation of more complex functions, such as the basic class of symmetric orthogonal polynomials (BCSOP) introduced in [18].