Generating asymptotics for factorially divergent sequences

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring is also closed under composition and inversion of power series. An 'asymptotic derivation' is defined which maps a power series to the asymptotic expansion of its coefficients. Product and chain rules for this derivation are deduced. With these rules asymptotic expansions of the coefficients of implicitly defined power series can be obtained. The full asymptotic expansions of the number of connected chord diagrams and the number of simple permutations are given as examples.


Introduction
This article is concerned with sequences a n , which admit an asymptotic expansion of the form, a n ∼ α n+β Γ(n + β) c 0 + c 1 α(n + β − 1) + c 2 α 2 (n + β − 1)(n + β − 2) + . . . , (1.1) for some α, β ∈ R >0 and c k ∈ R. Sequences of this type appear in many enumeration problems, which deal with coefficients of factorial growth. For instance, generating functions of subclasses of permutations and graphs of fixed valence show this behaviour [1,7]. Furthermore, there are countless examples where perturbative expansions of physical quantities admit asymptotic expansions of this kind [4,22,15]. The restriction to this specific class of power series is inspired by the work of Bender [6]. In this work the asymptotic behaviour of the composition of a power series, which has mildly growing coefficients, with a power series, which has rapidly growing coefficients, is analyzed. Bender's results are extended into a complete algebraic framework. This is achieved making heavy use of generating functions in the spirit of the analytic combinatorics or 'generatingfunctionology' approach [18,29]. The key step in this direction is to interpret the coefficients of the asymptotic expansion as another power series.
These structures bear many resemblances to the theory of resurgence, which was established by Jean Ecalle [16]. Resurgence assigns a special role to power series which diverge factorially, as they offer themselves to be Borel transformed. Jean Ecalle's theory can be used to assign a unique function to a factorially divergent series. This function could be interpreted as the series' generating function. Moreover, resurgence provides a promising approach to cope with divergent perturbative expansions in physics. Its application to these problems is an active field of research [15,2].
During a conversation with David Sauzin it became plausible that the presented methods can also be derived from resurgence. In fact, the formalism can be seen as a toy model of resurgence's calcul différentielétranger [16, Vol. 1] also called alien calculus [23,II.6]. This toy model is unable to fully reconstruct functions from asymptotic expansions, but does not rely on analytic properties of Borel transformed functions and therefore offers itself for combinatorial applications. A detailed and illuminating account on resurgence theory is given in David Sauzin's review [23,Part II] or [24]. For a review focused on applications to problems from physics consult [2].

Statement of results
Power series with well-behaved asymptotic expansions, as in eq. ( ], can be defined which maps a power series to the asymptotic expansion of its coefficients. A natural way to define such a map is to associate the power series n=0 c n x n to the series n=0 a n x n both related as in eq. (1.1). This map turns out to be a derivation, that means it fulfills a Leibniz rule and a chain rule, ). These statements will be derived from elementary properties of the Gamma function. Note that the chain rule involves a peculiar correction term if f has a non-trivial asymptotic expansion. The fact that the chain rule cannot be simple, that means for general f, g: , is obvious. The reasonable requirement that the function g(x) = x has a trivial asymptotic expansion, (A α β g)(x) = 0, would otherwise imply that all f ∈ R[[x]] α β have trivial asymptotic expansions. The formalism can be applied to calculate the asymptotic expansions of implicitly defined power series. This procedure is similar to the extraction of the derivative of an implicitly defined function using the implicit function theorem. In sections 2-4 the derivation ring R[[x]] α β will be described and the main Theorem 4.4, which establishes the chain rule for the asymptotic derivation, will be proven. In section 5, the apparatus will be applied to the calculation of the full asymptotic expansions of the number of connected chord diagrams and of the number of simple permutations.

Notation
] will be denoted in the usual 'functional' notation f (x) = ∞ n=0 f n x n . The coefficients of a power series f will be expressed by the same symbol with the index attached as a subscript f n or with the coefficient extraction operator [x n ]f (x) = f n . Ordinary (formal) derivatives are expressed as f ′ (x) = ∞ n=0 nf n x n−1 . The ring of power series, restricted to expansions of functions which are analytic at the origin, or equivalently power series with non-vanishing radius of convergence, will be denoted as R{x}. The O-notation will be used: O(a n ) denotes the set of all sequences b n such that lim sup n→∞ | bn an | < ∞ and o(a n ) denotes all sequences b n such that lim n→∞ bn an = 0. Equations of the form a n = b n + O(c n ) are to be interpreted as statements a n − b n ∈ O(c n ) as usual. See [5] for an introduction to this notation. Tuples of non-negative integers will be denoted by bold letters t = (t 1 , . . . , t L ) ∈ N L 0 . The notation |t| will be used as a short form for L l=1 t l .

Prerequisites
The first step is to establishing a suitable notion of a power series with a well-behaved asymptotic expansion.
]. Remark 2.3. The expression in eq. (2.1) represents an asymptotic expansion or Poincaré expansion with the asymptotic scale α n Γ(n + β) [14,Ch. 1.5]. Remark 2.4. For fixed R, an expansion as above with R explicit summands will be called an asymptotic expansion up to order R − 1.
β includes all (real) power series whose coefficients only grow exponentially: β . Remark 2.6. These with other series, which are in o(α n Γ(n + β − R)) for all fixed R ≥ 0, have an asymptotic expansion of the form in eq. (2.1) with all the c f k = 0. Remark 2.7. Definition 2.1 implies that f n ∈ O (α n Γ(n + β)). Accordingly, the power series in R[[x]] α β are a subset of Gevrey-1 sequences [20, Ch XI-2]. Being Gevrey-1 is not sufficient for a power series to be in R[[x]] α β . For instance, a sequence which behaves for large n as a n ∼ n!(1 Remark 2.8. In resurgence theory further restrictions on the allowed power series are imposed, which ensure that the Borel transformations of the sequences have proper analytic continuations or are 'endless continuable' [23,II.6]. These restrictions are analogous to the requirement that, apart from f n , also c k has to have a well-behaved asymptotic expansion. The coefficients of this asymptotic expansion are also required to have a well-behaved asymptotic expansion and so on. These kind of restrictions will not be necessary for the presented algebraic considerations, which are aimed at combinatorial applications. The central theme of this article is to interpret the coefficients c f k of the asymptotic expansion as another power series. In fact, Definition 2.1 immediately suggests to define the following map: ] be the map that associates a power series Corollary 2.10. A α β is linear. Remark 2.11. In the realm of resurgence such an operator is called alien derivative or alien operator [23, II.6]. 2) is normalized such that shifts in n can be absorbed by shifts in β. More specifically, Proof. Suppose the first m coefficients of f vanish. Set g(x) = f (x) x m or g n = f n+m . Eq. (2.2) gives, ] α β with β ≤ 0 is to use the field of (formal) Laurent series R((x)) as the target space for A α β and demand that negative powers of x commute with the A α β operator.

A derivation for asymptotics
The following lemma forms the foundation for most conclusions in this article. It provides an estimate for sums over Γ functions. Moreover, it ensures that the subspace R[[x]] α β of formal power series falls into a large class of sequences studied by Bender [6]. From another perspective the lemma can be seen as an entry point to resurgence, which bypasses the necessity for analytic continuations and Borel transformations. . This way, the sum n m=0 G n m can be estimated after stripping off the two boundary terms: It follows from nΓ(n) = Γ(n + 1) that G n 1 = G n 0 β n−1+β for all n ≥ 1. Substituting this into eq. (3.2) gives the estimate in eq. (3.1) with C = (2 + β)Γ(β). The remaining case n = 0 is trivial.
Proof. Rewrite the left hand side as . Lemma 3.1 can be applied with the substitutions β → β + R, n → n − 2R to obtain the required estimate.
Proof. This inequality is merely an iterated version of Lemma 3.1. It can be proven by induction in L. The case L = 1 is trivial. Lemma 3.1 guarantees the existence of a C ∈ R such that Suppose the statement holds for L. Using the statement on C L Γ(n − m L+1 + β) on the left hand side results in which is the statement for L + 1.

An immediate consequence of Corollary 3.2 is
β is a derivation, that means it fulfills the Leibniz rule . The coefficients h n are given as a sum by the Cauchy product formula. This sum can be written suggestively as Definition 2.9 guarantees that the first two sums have sound asymptotic expansions for large n. Together they constitute an asymptotic expansion of h n up to order R − 1. We verify this exemplary on the first sum, where the asymptotic expansion from eq. (2.2) up to order R − m − 1 of f n−m can be substituted: . It remains to be shown that the last sum in eq. (3.6) is negligible. Because f n , g n ∈ O(α n Γ(n + β)), there is a constant C such that |f n | ≤ Cα n Γ(n + β) and |g n | ≤ Cα n Γ(n + β) for all n ≥ 0. Hence, shows that this sum is in O(α n Γ(n − R + β)) by Corollary 3.2.
Proof. Proof by induction on L using the Leibniz rule.
Although the last three statements are only basic general properties of commutative derivation rings, they suggest that A α β fulfills a simple chain rule. In fact, Corollary 3.8 can still be generalized from polynomials to analytic functions, but, as already mentioned, this intuition turns out to be false in general.

Composition by analytic functions
In [6] Edward Bender established this theorem for the case L = 1 in a less 'generatingfunctionology' biased notation. If for example g ∈ R[[x]] α β and f ∈ R{x, y}, then his Theorem 1 allows us to calculate the asymptotics of the power series f (g(x), x). In fact, Bender analyzed more general power series including sequences with even more rapid than factorial growth.
The following proof of Theorem 4.1 is a straightforward generalization of Bender's Lemma 2 and Theorem 1 in [6] to the multivariate case f ∈ R{y 1 , . . . , y L }.
Proof. The proof is a straightforward application of Corollary 3.4. There is a constant C such that g l n ≤ Cα n Γ(n + β) for all n ∈ N 0 and l ∈ [1, L]. Accordingly, An application of Corollary 3.4 results in the lemma.
Proof. As a consequence of Proposition 2.13, g l (x) Proof of Theorem 4.1. The composition of two power series can be expressed as the sum which can be split in preparation for the extraction of asymptotics: The left sum is just the composition by a polynomial. Corollary 3.8 asserts that this sum is in R[[x]] α β . It has the asymptotic expansion given in eq. (3.9) which agrees with the right hand side of eq. (4.1) up to order R − 1, because the partial derivative reduces the order of a polynomial by one and g l = 0. It is left to prove that the coefficients of the power series given by the right sum are in O(α n Γ(n − R + β)). Corollary 4.3 and the fact that there is a constant C, such that |f t1,...,tL | ≤ C |t| for all t ∈ N L 0 , due to the analyticity of f , ensure that there is a constant C ′ ∈ R such that for all n ≥ R + 1. The result of the last sum |{t 1 , . . . , t L ∈ N 0 |t 1 + . . . + t L = t}| = t+L−1 L−1 is a polynomial in t. Corollary 3.3 asserts that the remainder sum is in O (α n Γ(n + β − R)).

Proof of the main theorem: Composition of power series in R[[x]] α β
Despite the fact that Bender's theorem applies to a broader range of compositions f • g, where f does not need to be analytic and g does not need to be an element of The problem is that we cannot truncate the sum ∞ k=0 f k g(x) k without losing significant information. In this section, this obstacle will be confronted and the general chain rule for the asymptotic derivative will be proven. Let 1 Julien Courtiel remarked that the statement f ∈ Diff id (R, 0) α β ⇒ f −1 ∈ Diff id (R, 0) α β was not obvious in a previous version of this manuscript. The argument in this version was modified to be more transparent in this respect.
The tail of the sum in eq. (4.6) over m turns out to be asymptotically negligible. However, in contrast to the preceding arguments, the sum cannot be truncated at a fixed value of m independent of n. A cutoff that grows slowly with n has to be introduced. More specifically, Lemma 4.6. If f, g, A, B as above, then there is a constant C ∈ R such that where s(n) = ⌈4(R log n + C)⌉.
Proof. The proof proceeds by substitution of the asymptotic expansion up to order R − 1 of B(x)A(x) m into eq. (4.7). This can be done, because n−m is large. The result is [ where the rest term, O(α n Γ(n + β + 2 − R)), was omitted. An elementary variant of the Chu-Vandermonde identity (Lemma B.2) can be used to expand the product of the binomial and Γ-function. This expansion allows us to perform the summation over m behind the coefficient extraction: The sum over l can be truncated at order R − 1 − k, because the summands can be estimated by C l P (l)α n Γ(n + β + 2 − k − l) with some C ∈ R and P ∈ R[l] each depending on k. Corollary 3.3 asserts that the truncated part is in O(α n Γ(n + β + 2 − R)). The subsequent change in summation variables k → k + l gives rise to, which results in the statement after noting that the sums over l and m can be completed, because lim n→∞ s(n) = ∞.
Proof of Theorem 4.4. The rest of the proof is merely an algebraic exercise. We start with the expression from Lemma 4.7 for [x k ] A α β+2 f • g −1 (x) and use the chain and product rules from Proposition 3.5, Theorem 4.1 as well as .
Remark 4.8. Bender and Richmond [8] established that [x n ](1 + g(x)) γn+δ = nγe γg 1 α g n + O(g n ) if g n ∼ αng n−1 and g 0 = 0. Using Lagrange inversion the first coefficient in the expansion of the compositional inverse in eq. (4.5) can be obtained from this. In this way, Theorem 4.4 is a generalization of Bender and Richmond's result. In the same article Bender and Richmond proved a theorem similar to Theorem 4.4 for the class of power series f which grow more rapidly than factorial such that nf n−1 ∈ o(f n ). Theorem 4.4 establishes a link to the excluded case nf n−1 = O(f n ).

Connected chord diagrams
A chord diagram with n-chords is a circle with 2n points, which are labeled by integers 1, . . . , 2n and connected in disjoint pairs by n-chords. There are (2n − 1)!! of such diagrams. A chord diagram is connected if no set of chords can be separated from the remaining chords by a line which does not cross any chords. Let I(x) = n=0 (2n − 1)!!x n and C(x) = n=0 C n x n , where C n is the number of connected chord diagrams with n chords. Following [17], the power series I(x) and C(x) are related by, This functional equation can be solved for the coefficients of C(x) by basic iterative methods. The first few terms are, This sequence is entry A000699 in Neil Sloane's integer sequence on-line encyclopedia [26]. (1 + C(xI(x) 2 )) = A 2 which can be solved for (A 2 1 2 C)(x) and simplified using eq. (5.1), A further simplification can be achieved by utilizing the linear differential equation 2x 2 I ′ (x) + xI(x)+1 = I(x) from which the differential equation [17] can be deduced:  This is the generating function of the full asymptotic expansion of C n . The first few terms are, Expressed in the traditional way using eq. (2.2) from Definition 2.9 this becomes The first term, e −1 , in this expansion has been computed by Kleitman [21], Stein and Everett [28] and Bender and Richmond [8] each using different methods. With the presented method an arbitrary number of coefficients can be computed. Some additional coefficients are given in Table 1.
The probability of a random chord diagram with n chords to be connected is therefore e −1 (1−

Monolithic chord diagrams
A chord diagram is called monolithic if it consists only of a connected component and of isolated chords which do not 'contain' each other [17]. That means with (a, b) and (c, d) the labels of two chords, it is not allowed that a < c < d < b and c < a < b < d. Let M (x) = n=0 M n x n be the generating function of monolithic chord diagrams. Following [17], M (x) fulfills C)(x) in eq. (5.5) gives Some additional coefficients are given in Table 1. The probability of a random chord diagram with n chords to be non-monolithic is therefore  |x n , the generating function of simple permutations, and F (x) = ∞ n=1 n!x n , the generating function of all permutations. Following [1], S(x) and F (x) are related by the equation,
As F (x) ∈ R[[x]] 1 1 and (A 1 1 F ) = 1, the full asymptotic expansion of S(x) can be obtained by applying the chain rule (Theorem 4.4) to both side of eq. (5.10). Alternatively, eq. (5.10) implies x−x 2 1+x = 0 together with the expression for the asymptotic expansion of F −1 (x) in terms of (A 1 1 F )(x) from eq. (4.5) shows that, This can be reexpressed using the functional equation (5.10), F −1 (F (x)) = x as well as the differential equation The coefficients of (A 1 1 S)(x) can be computed iteratively. The first few terms are up to order n is as easy as calculating the expansion of S(x) or F −1 (x) up to order n + 2. Some additional coefficients are given in Table 2. Applications include functional equations for 'irreducible combinatorial objects'. The two examples fall into this category. Irreducible combinatorial objects were studied in general by Beissinger [3].
A Some remarks on differential equations Differential equations arising from physical systems form an active field of research in the scope of resurgence [19,25]. A detailed exposition of the application of resurgence theory to differential equations can be found in [13]. Unfortunately, the exact calculation of an overall factor of the asymptotic expansion of a solution of an ODE, called Stokes constant, turns out to be difficult for many problems. This fact severely limits the utility of the method for enumeration problems, as the dominant factor of the asymptotic expansion is of most interest and the detailed structure of the asymptotic expansion is secondary.
In this section it will be sketched, for the sake of completeness, how the presented combinatorial framework fits into the realm of differential equations. The given elementary properties each have their counterpart in resurgence's alien calculus [23,II.6].
Theorem 4.1 serves as a good starting point to analyze differential equations with power series solutions in R[[x]] α β . Given an analytic function F ∈ R{x, y 0 , . . . , y L }, the A α β -derivation can be applied on the ordinary differential equation The differential equation becomes a linear equation for the asymptotic expansions of the derivatives f (l) . This raises the question how these different asymptotic expansions relate to each other.
Proof. The statements can be verified by using f ′ (x) = ∞ n=0 nf n x n−1 and substituting an asymptotic expansion up to order R − 1 from eq. (2.2). Set h(x) = x 2 f ′ (x) such that, for n ≥ 1