A Further Property of Functions in Class B(m): An Application of Bell Polynomials

We say that a function α(x) belongs to the set A(γ) if it has an asymptotic expansion of the form α(x) ∼ ∑∞i=0 αix as x→ ∞, which can be differentiated term by term infinitely many times. A function f (x) is in the class B(m) if it satisfies a linear homogeneous differential equation of the form f (x) = ∑m k=1 pk(x) f (k)(x), with pk ∈ A(ik), ik being integers satisfying ik ≤ k. These functions appear in many problems of applied mathematics and other scientific disciplines. They have been shown to have many interesting properties, and their integrals ∫ ∞ 0 f (x) dx, whether convergent or divergent, can be evaluated very efficiently via the Levin–Sidi D(m)-transformation, a most effective convergence acceleration method. (In case of divergence, these integrals are defined in some summability sense, such as Abel summability or Hadamard finite part or a mixture of these two.) In this note, we show that if f (x) is in B(m), then so is ( f ◦ g)(x) = f (g(x)), where g(x) > 0 for all large x and g ∈ A(s), s being a positive integer. This enlarges the scope of the D(m)-transformation considerably to include functions of complicated arguments. We demonstrate the validity of our result with an application of the D(3) transformation to two integrals I[ f ] and I[ f ◦ g], for some f ∈ B(3) and g ∈ A(2). The Faà di Bruno formula and Bell polynomials play a central role in our study.


Introduction and Main Result
The D (m) transformation is a very effective convergence acceleration tool for computing infinite-range integrals of the form ∫ ∞ 0 f (x) dx, whose integrands f (x) belong to the function class B (m) , m being a positive integer. Both the D (m) transformation and the function class B (m) were introduced by (Levin & Sidi, 1981) and studied further in (Sidi, 2003, Chapter 5). Most special functions appearing in applied mathematics and most functions arising in different scientific and engineering disciplines belong to the sets B (m) . Since it is clear that, as methods of convergence acceleration, the D (m) transformations, m = 1, 2, . . . , have a very large and ever increasing scope. To date, these transformations are the most effective means for computing infinite-range integralswhether convergent or divergent-of functions in the classes B (m) . 1

The Function Class A (γ)
Before recalling the definition of the class B (m) , we recall the definition of another function class that was introduced and denoted A (γ) also in (Levin & Sidi, 1981). The classes A (γ) feature prominently in the definition of the class B (m) , as will be clear soon.
Definition 1.1. A function α(x) belongs to the set A (γ) , where γ is complex in general, if it is infinitely differentiable for all large x > 0 and has a Poincaré-type asymptotic expansion of the form (1.1) and its derivatives have Poincaré-type asymptotic expansions obtained by differentiating that in (1.1) formally term by term.
Remarks A. The following are simple consequences of Definition 1.1. We shall make use of them later. For more, see (Sidi, 2003, Chapter 5).
, then, for any positive integer k, α ∈ A (γ+k) but not strictly. Conversely, if α ∈ A (δ) but not strictly, then α ∈ A (δ−k) strictly for a unique positive integer k.
(Concerning the uniqueness of α(x), see the last paragraph of (Sidi, 2003, Appendix A).) To avoid this, in certain places, it is more convenient to work with subsets X (γ) of A (γ) , which are defined next.
Definition 1.2. The subsets X (γ) of A (γ) are defined for all γ collectively as follows: is closed under addition and multiplication by scalars.
Functions α(x) that are given as sums of series ∑ ∞ i=0 α i x γ−i that converge for all large x form a subset of X (γ) ; obviously, such functions are of the form α(x) = x γ R(x) with R(x) analytic at infinity. Thus, R(x) can be rational functions that are bounded at infinity, for example.

The Function Class B (m)
We now turn to the definition of the class B (m) .

Definition 1.3. A function f (x) that is infinitely differentiable for all large x belongs to the set B (m) if it satisfies a linear homogeneous ordinary differential equation of order m of the form
where either p k ∼ 0 or p k ∈ A (i k ) strictly for some integer i k ≤ k, 1 ≤ k ≤ m − 1, and p m ∈ A (i m ) strictly for some integer i m ≤ m.
Remarks B. The following are consequences of Definition 1.1. They can be found in (Levin & Sidi, 1981) and (Sidi, 2003, Chapter 5).
B3. If f ∈ B (m) with smallest m, then the differential equation (1.2) satisfied by f (x) is unique, provided the p k are restricted (to X (k) instead of A (k) ) such that either p k ≡ 0 or p k ∈ X (i k ) strictly for some integer i k ≤ k, 1 ≤ k ≤ m, and p m ∈ X (i m ) strictly for some integer i m ≤ m. See (Sidi, 2003, p. 99, Proposition 5.1.5). B4 . . , µ, then the following are true: 2 B6. If g i ∈ B (r) , i = 1, . . . , µ, and satisfy the same ordinary differential equation, then the following are true: ) . m) and is integrable at infinity, then, under some additional minor conditions at x = ∞, where ρ k are integers depending only on the p k (x) and satisfy This result forms the basis of the D (m) transformation of (Levin & Sidi, 1981), which has proved to be an extremely efficient convergence accelerator for the computation of the integrals ∫ ∞ 0 f (x) dx, as mentioned in the beginning of this section.
By Remarks B1, B2, B5, and B6, it is clear that the classes B (m) contain an ever increasing number of functions with varying behavior (oscillatory or nonoscillatory or combinations of the two), and this implies that the D (m) transformation is a comprehensive convergence acceleration method with ever increasing scope.
Finally, we would like to mention again that most special functions that appear in scientific and engineering applications belong to one of the classes B (m) .

Main Results
In this note, we continue our exploration of the properties of the classes B (m) . Analogous to what happens to the sum f + g and the product f g of two functions f and g, discussed in Remarks B5 and B6 above, we wish to explore what happens to their composition. Specifically, we address the following question: We answer this question in Theorem 1.5, which follows as a corollary from Theorem 1.4; both theorems are the main results of this note. We provide the proofs of these theorems in the next section, where we make repeated use of Remarks A1-A8 without mentioning them. Finally, to keep the proofs simpler, we replace the sets A (γ) by their subsets X (γ) , even though the assertions of Theorems 1.4 and 1.5 are true with the sets A (γ) . We also note that the Faà di Bruno formula and Bell polynomials play a central role in our proof.
Theorem 1.4. Let f (x) be a solution to the linear homogeneous differential equation of order m Let also g ∈ X (s) strictly for some positive integer s, such that lim x→∞ g(x) = +∞. Then ϕ(x) ≡ f (g(x)) satisfies a linear homogeneous differential equation of order m of the form where the π k are determined by the p k and are such that either π k ≡ 0 or π k ∈ X (r k ) strictly for some integer r k , 1 ≤ k ≤ m − 1, and π m ∈ X (r m ) strictly for some integer r m . Actually, we have and (1.8) (Note: On the right-hand side of the inequality in (1.8), s(i k − k) is absent when p k ≡ 0, and r n − n is absent when π n ≡ 0 for n ∈ {k + 1, k + 2, . . . , m − 1}.) In addition, (1.9) (The explicit expression for π m is given in (2.16). The rest of the π k are given by the recursion relation in (2.18).) Theorem 1.5. Let f (x) be in B (m) and let g(x) be in X (s) strictly for some positive integer s, such that lim x→∞ g(x) = +∞.
Clearly, Theorem 1.5 expands considerably the scope of the class B (m) , hence the scope of the D (m) transformation, to include functions of complicated arguments, in the following sense: If the D (m) transformation accelerates the convergence of the integral ∫ ∞ 0 f (x) dx, it also accelerates the convergence of the integral More generally, the D (m ′ ) transformation, for some m ′ , accelerates the convergence of the integral ) dx when g ∈ X (s) strictly for some positive integer s and h(x) is an arbitrary function in B (r) for some r. (This follows from Remark B5 with m ′ ≤ m + r.) In Section 3, we demonstrate the validity of our result with an application of the D (3) transformation to two integrals I [ f ] and I [ f • g], for some f ∈ B (3) and g ∈ A (2) .
In Section 4, we show via an example that f ∈ B (m) with minimal m does not necessarily mean that m is minimal also for ϕ(x) even though ϕ ∈ B (m) too by Theorem 1.5 .

Preliminaries
First, we note that, being in X (s) , g(x) has an asymptotic expansion of the form g n x s−n as x → ∞ and g 0 > 0, (2.1) from which, we also have that is a polynomial of degree s or behaves like one as x → ∞.
Thus, using the notation we also have that and, since ( ∑ s n=0 g n x s−n ) (i) = 0 for i ≥ s + 1, where g s+µ , µ ≥ 1, is the first nonzero g s+ j with j ≥ 1, assuming that g (i) 0. Of course, g s+ j = 0, j = 1, 2, . . . , ⇒ g (i) (x) ≡ 0, i = s + 1, s + 2, . . . , and this can occur when g(x) = ∑ s n=0 g n x s−n , for example, in which case, g (i) (x) ≡ 0 for i = s + 1, s + 2, . . . . Summarizing, we have Next, it is clear that we need to prove that ϕ(x) = f (g(x)) satisfies (1.6), with π k ∈ X (r k ) for some integer r k when π k 0. Replacing x by g(x) throughout the differential equation (1.5) satisfied by f (x), we have and this is the starting point of our proof and is most important. Here, we emphasize that f (k) (g(x)) stands for the kth derivative of f with respect to its argument, evaluated at g(x); that is, (x) . Thus, f (k) (g(x)) does not stand for the kth derivative of ϕ(x) = f (g(x)) with respect to x.
Whenever convenient, in the sequel, we write p k , π k , and g (i) instead of p k (x), π k (x), and g (i) (x), respectively, for short. Thus, p k (g) and f (k) (g) stand for p k (g(x)) and f (k) (g(x)), respectively.

Special Cases
Before embarking on the proof for arbitrary m, we look at the simple but instructive cases involving m = 1, 2.

The Case m = 1
Here we consider two different cases.
Note that if p 1 ≡ 0, the term s(i 1 − 1) is absent throughout.

The Case of Arbitrary m
We prove Theorem 1.4 first. We start with (2.6). By the Faà di Bruno formula (see (Faà di Bruno, 1855, 1857) for differentiation of f • g, we have where B n,k (y 1 , y 2 , . . . , y n−k+1 ) is the Bell polynomial (see (Bell, 1934)) defined as in the summation being on the nonnegative integers j 1 , j 2 , . . . , j n−k+1 such that (2.9) (The simplest of these polynomials are B n,1 (y 1 , . . . , y n ) = y n and B n,n (y 1 ) = y n 1 .) For a detailed treatment of the Faà di Bruno formula and related topics, see the excellent review by (Johnson, 2002). See also (Roman, 1980), for example. For Bell polynomials, see also (Roman,1984). Now, if the conjecture in (1.6) is true, then substituting (2.7) in (1.6), we must have , (2.10) which, upon changing the order of summation, becomes ).
By the fact that B n,n (y 1 ) = y n 1 , and writing g (i) instead of g (i) (x) for short, we have Since g ′ (x) > 0 for all large x, this diagonal is positive, hence the linear system in (2.12) has a unique solution for π 1 (x), . . . , π m (x). With the existence of the π k established, we now need to show that, π k ∈ X (r k ) strictly for some integer r k when π k 0. We achieve this goal by induction on k, in the order k = m, m − 1, . . . , 2, 1.
( 2.15) We now start the induction with π m , which we obtain from the last of the equations in (2.12). Thus, (2.16) By the fact that p m (g) ∈ X (si m ) strictly and (g ′ ) m ∈ X ((s−1)m) strictly, it is clear that π m 0 and π m ∈ X (r m ) strictly, r m = si m − m(s − 1) = s(i m − m) + m. (2.17) We have thus shown the validity of our assertion for π m .
We now continue by induction on k. Let us assume that the assertion is true also for π m−1 , π m−2 , . . . , π k+1 , namely, π n ∈ X (r n ) strictly for some integer r n if π n 0, n ∈ {m − 1, m − 2, . . . , k + 2, k + 1}. The proof will be complete if we show that π k ∈ X (r k ) strictly for some integer r k if π k 0. Solving (2.12), namely, p k (g) = ∑ m n=k π n L n,k , for π k , we obtain First, L k,k ∈ X (sk−k) strictly, and p k (g) ∈ X (si k ) strictly when p k 0; therefore, if p k 0, If p k ≡ 0, then p k (g) ≡ 0 too, and, therefore, p k (g)/L k,k ≡ 0.