Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian

We introduce the notion of controlled Loewner-Kufarev equations and discuss aspects of the algebraic nature of the equation embedded into the (Sato)-Segal-Wilson Grassmannian. Further, we relate it to conformal field theory (CFT) and free probability.


Introduction
C. Loewner [24] and P. P. Kufarev [25] initiated a theory which was then further extended by C. Pommerenke [31], which shows that given any continuously increasing family of simply connected domains containing the origin in the complex plane, the inverses of the Riemann mappings associated to the domains are described by a partial differential equation, the so-called Loewner- (Kufarev) equation (see Section 2.2 for details). More recently, I. Markina and A. Vasil'ev [27,29] considered the so-called alternate Loewner-Kufarev equation, which describes not necessarily increasing chains of domains.
In this paper, we further generalise it and introduce a class of controlled Loewner-Kufarev equations, df t (z) = zf ′ t (z){dx 0 (t) + dξ(x, z) t }, f 0 (z) ≡ z ∈ D 0 , where D 0 is the unit disc in the complex plane centred at zero, x 0 , x 1 , x 2 , · · · are given functions which will be called the driving functions, x = (x 1 , x 2 , · · · ) and ξ(x, z) t = ∞ n=1 x n (t)z n . In writing down the controlled Loewner-Kufarev equation, the right hand side becomes − ∞ n=0 (L n f )(z)dx n , where L n = −z n+1 ∂/(∂z) for n ∈ Z, the L n are exactly the generators of the non-negative part of the Witt algebra, or put differently, central charge zero Virasoro algebra. Therefore, we are going to consider an extension of [9], where the second author established and studied the role of Lie vector fields, boundary variations and the Witt algebra in connection with the Loewner-Kufarev equation.
So, let us start first with the classical work of A. A. Kirillov and D. V. Yuriev [17] / G.B. Segal and G. Wilson [34] / N. Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada [16] which will be also fundamental and basic in the present context, in particular in understanding the appearance of the Virasoro algebra with nontrivial central charge.
A. A. Kirillov and D. V. Yuriev [17], constructed a highest weight representation of the Virasoro algebra, where the representation space is given by the space of all holomorphic sections of an analytic line bundle over the orientation-preserving diffeomorphism group Diff + S 1 of the unit circle S 1 (modulo rotations). They also gave an embedding of (Diff + S 1 )/S 1 into the infinite dimensional Grassmannian. In fact, this embedding is an example of a construction of solutions to the KdV hierarchy found by I.Krichever [21], which we address in Section 3.2.
If we embed a univalent function on the unit disc D 0 into the infinite dimensional Grassmannian, by the methods of Kirillov-Yuriev [17], Krichever [21], or Segal-Wilson [34], then one needs to track the Faber polynomials and Grunsky coefficients associated to the univalent function. In general, it is not easy to calculate them from the definition.
One of our main results is, however, the following. R. Friedrich [9] proposed to lift the embedded Loewner-Kufarev equation to the determinant line bundle over the Sato-Segal-Wilson Grassmannian Gr(H), as a natural extension of the "Virasoro Uniformisation" approach by M. Kontsevich [18] / R. Friedrich and J. Kalkkinen [8] to construct generalised stochastic / Schramm-Loewner evolutions [33] on arbitrary Riemann surfaces, which would also yield a connection with conformal field theory in the spirit of [34,16]. Let us also mention the work of B. Doyon [3], who uses conformal loop ensembles (CLE), which can be nicely related with the content of the present article.
In [29], I. Markina and A. Vasil'ev established basic parts of this program, by considering embedded solutions to the Loewner-Kufarev equation into the Segal-Wilson Grassmannian and related the dynamics therein with the representation of the Virasoro algebra, as discussed by Kirillov-Yuriev [17].
Further, they considered the tau-function associated to the embedded solution as a lift to the determinant line bundle.
As observed and briefly discussed in [18,8], the generator of the stochastic Loewner equation is hypo-elliptic.
This observation was more recently worked out and detailed by J. Dubédat [4,5]. Also, I. Markina, I. Prokhorov and A. Vasil'ev [26] observed and discussed the sub-Riemannian nature of the coefficients of univalent functions. This connects nicely with the general theory of hypo-elliptic flows, as explained e.g. in the book by F. Baudoin [1], and which led to propose a connection of the global geometry of (stochastic) Loewner-Kufarev equations with rough paths. Now, in the theory of rough paths (see e.g., the introduction in [23]), one of the central objects of consideration is the following controlled differential equation: where X t is a continuous path in a normed space V , called the input of (1.1). On the other hand, the path Y t is called the output of (1.1). When we deal with this equation, an important object is the signature of the input X t , with values in the (extended) tensor algebra associated with V and which is written in the following form: S(X) s,t := (1, X 1 s,t , X 2 s,t , · · · , X n s,t , · · · ), s t. If X t has finite variation with respect to t, then each X n s,t is the nth iterated integral of X t over the interval [s, t]. With this object, a combination of the Magnus expansion and the Chen-Strichartz expansion theorem (see e.g., [1,Section 1.3]) tells us that the output Y t is given as the result of the action of S(X) 0,t applied to Y 0 .
Heuristically, we may say that a 'group element' S(X) in some big 'group' acts on some element in the (extended) tensor algebra T ((V )) which gives the output Y t , or it might be better to say that the vector field ϕ defines how the 'group element' acts on the algebra. In this spirit, we would like to describe such a picture in the context of controlled Loewner-Kufarev equations.
For this, we extract the algebraic structure of the controlled Loewner-Kufarev equation. If we regard the driving functions x 0 , x 1 , x 2 , · · · just as letters in an alphabet then it turns out that explicit expressions for the associated Grunsky coefficients are given by the algebra of formal power series, where the space of coefficients is given by words over this alphabet. It is worth mentioning that the action of the words over this alphabet will be actually given by the negative part of the Witt generators. Thus the action of the signature encodes many actions of such negative generators. This can be used to derive a formula for f t (z) as the signature 'applied' to the initial data f 0 (z) ≡ z (see Theorem 3.8).
However, the story so far lets us ask how the signature associated with the driving functions describes the corresponding tau-function rather than the f t itself.
Theorem 1.2 (see Theorem 3.9). Along the solution of the controlled Loewner-Kufarev equation, the associated tau-function can be written as the determinant of a quadratic form of the signature.
Let us now summarise the structure of the paper. In Section 2, we formulate solutions f t (z) to controlled Loewner-Kufarev equations. We add also a brief review of the classical Loewner-Kufarev equation, and then explain how the classical one is recovered by the controlled Loewner-Kufarev equation. We track the variation of the Taylor-coefficients of f t and also the Faber polynomials and Grunsky coefficients. In Section 3, we first recall briefly basics of the Segal-Wilson Grassmannian and Krichever's construction. After that, we describe how a univalent function on D 0 is embedded into the Grassmannian. We extract the algebraic structure of the controlled Loewner-Kufarev equations in order to obtain Theorem 3.9. Finally, in Section 4, we establish and briefly discuss connections of the material considered so far with CFT and free probability. All fundamental things about Faber polynomials and Grunsky coefficients are wrapped up in Appendix A. Several proofs of propositions are also relegated to Appendix A, because they tend to be rather long (in particular, the proof of Proposition 2.9 will be very long) as we intended not to interrupt the flow of the principal story.

Controlled Loewner-Kufarev equation
2.1. Definition of solutions to controlled Loewner-Kufarev equations. Given functions x 1 , x 2 , · · · : [0, T ] → C, we will write x := (x 1 , x 2 , · · · ) and x n (t)z n for z ∈ C if it converges. If A : [0, T ] → C is of bounded variation, we write dA or A(dt) (when emphasising the coordinate t on [0, T ]) the associated complex-valued Lebesgue-Stieltjes measure on [0, T ], and the total variation measure will be denoted by |dA|.
Definition 2.1. Let T > 0. Suppose that x 0 : [0, T ] → R, x 1 , x 2 , · · · : [0, T ] → C are continuous and of bounded variations, and x 0 (0) = 0. Let f t : D 0 → C be conformal mappings for 0 t T . We say {f t } 0 t T is a solution to (2) ∞ n=1 n [0,T ] |dx n |(t)z n converges uniformly in z on each compact set in D 0 ; is continuous with respect to the uniform norm on K; (4) it holds that In the sequel, we refer to equation (2.1) as a controlled Loewner-Kufarev equation (with driving paths x 0 and x := (x 1 , x 2 , · · · )).
(a) We do not put the univalency condition into the notion of solution. However, the condition (1) and (3) Remark 2.2. Let {f t } 0 t T be a solution to the controlled Loewner-Kufarev equation (2.1). Then the following holds: (a) For each w ∈ C and a compact set K ⊂ D 0 , the mapping t → sup z∈K |f ′ t (z) − w| is continuous since we have and the condition in Definition 2.1-(3).
(b) The mapping t → f t is continuous with respect to the uniform norm on each compact set in D 0 . In fact, by using Definition 2.1-(4), we have for each compact where z 0 ∈ K is such that |z 0 | = sup z∈K |z|. By Definition 2.1-(3) (or the above remark (a)), we find that sup 0 u T f ′ u | K ∞ < +∞. Therefore, by using the dominated convergence theorem with Definition 2.1-(2), the right-hand side goes to zero as |t − s| → 0.  (1) f t is analytic and univalent on D 0 for each 0 t T ; (2) f t (z) = e t z + a 2 (t)z 2 + · · · for z ∈ D 0 ; (  T , the function f t (z) = e t z + · · · is analytic in |z| < r 0 , the mapping [0, T ] ∋ t → f t (z) is absolutely continuous for each |z| < r 0 , and |f t (z)| K 0 e t for all |z| < r 0 and t ∈ [0, T ].
for all |z| < r 0 and for almost all t ∈ [0, T ].
According to the terminology in [2] we call the equation (2.2) the Loewner-Kufarev equation (if we regard p(t, z) as given and f t (z) as unknown).
Because of the equation (2.2), it holds that p(t, 0) = lim z→0 ( ∂ ∂t f t (z))/(zf ′ t (z)) = 1, and hence the 'Herglotz Representation Theorem' applies, which permits to conclude that, for every t ∈ [0, T ], there exists a probability measure ν t on S 1 = ∂D 0 (which is naturally identified with [0, 2π] as measurable spaces, and then the induced probability measure is still denoted by ν t ) such that Substituting this into (2.2), the Loewner-Kufarev equation becomes Assuming that ν t (dθ) =: ν t (θ)dθ, is sufficiently regular, we write the Fourier series of ν t (θ) as We temporally introduce the notations x 0 (t) := t 0 a 0 (s)ds and for k = 1, 2, · · · . By using the relations 1 2π for k = 1, 2, · · · and |z| < 1, equation (2.3) assumes the following form: This can be rewritten as the following controlled differential equation If we omit the condition Re{p(t, z)} > 0, that is, we allow the real part of p(t, z) to have arbitrary signs, then the equation (2.2) is called alternate Loewner-Kufarev equation, as considered by I. Markina and A. Vasil'ev [27]. Intuitively, this describes evolutions of conformal mappings whose images of D 0 are not necessary increasing, i.e. not strict subordinations. It appears that the general theory with respect to the existence and uniqueness of solutions is not yet fully developed. However, our controlled Loewner-Kufarev equations (2.1) deal with this alternate case because we have not assumed that p(t, z) := d dt (x 0 (t) + ξ(x, z) t ) has a positive real part. Remark 2.3. Readers focusing on radial Loewner equations might feel puzzled by the heuristic assumption that the Radon-Nikodym density νt(dθ) dθ = ν t (θ) exists, because the radial Loewner equation describes the case ν t (dθ) = δ e iw(t) (dθ) where w(t) is a continuous path in R, so that there does not exist a Radon-Nikodym density. However, several explicit examples of Loewner-Kufarev equations within this setting, are presented with simulations in Sola [35].
By applying variation of constants to (2.7), we obtain the following recurence relation for n 2, and we get The proof is similar to the one of Proposition 2.12 below and hence omitted.

Variation of Grunsky coefficients induced by a Loewner-Kufarev equation.
Here is an open neighbourhood of the origin; is continuous and of bounded variation; as Lebesgue-Stieltjes measures on [0, T ].
The proof of Proposition 2.9 is based on classical techniques to prove inverse function theorems however a little bit involved. Therefore we shall postpone it to Appendix A.2.
Let {f t } 0≤t≤T be a solution to the controlled Loewner-Kufarev equation (2.1). Because of Corollary 2.7, associated to f t (z) are Faber polynomials and Grunsky coefficients (see Appendix A-Definition A.1) which will be denoted by Q n (t, w) and b −n,−m (t) respectively. Proposition 2.10.
(i) (Variation of Faber polynomials): We have for each n ∈ N, (ii) (Variation of Grunsky coefficients): For each n, m ∈ N, with the initial conditions b −n,−m (0) = 0 for all n, m ∈ N.
Proof. (i) Let n ∈ N. Let U and V be as in Proposition 2.9. Then f −1 t (ζ), ζ ∈ V satisfies the equation Let X 0 ⊂ V be an open disc centred at 0. By using Cauchy's integral formula, we have for w ∈ X 0 , Noting that the orientation of ∂X 0 is anti-clockwise, we get and hence the result.
(ii) Since f t (z) satisfies the controlled Loewner-Kufarev equation, by putting p(dt, z) : On the other hand, by Proposition 2.10-(i), we have from which we conclude (2.12) Combining (2.11) and (2.12), we obtain and then by comparing with (2.10), we get the result. The initial condition is obviously derived from f 0 (z) ≡ z.
To derive an explicit formula for general b −n,−m (t), we shall introduce some notation.
(2) Suppose that x 1 , x 2 , · · · , x p+q : [0, T ] → C are continuous and of bounded variations. Then for each 0 t T , we set The general formula for the Grunsky-coefficients along the controlled Loewner-Kufarev equation (2.1) is given by the following. The proof is given in Section A.3. Proposition 2.12. For n, m ∈ N and t 0, where, for m = i 1 + · · · + i p + r and n = j 1 + · · · + j q + s, we have put and w(r, s) i 1 ,··· ,ip;j 1 ,··· ,jq := w(r) i 1 ,··· ,ip;∅ w(s) ∅;j 1 ,··· ,jq .  (2) The orthogonal projection pr − : W → H − is compact. The Fredholm index of the orthogonal projection pr + : W → H + is called the virtual dimension of W . For d ∈ Z, we set If we take W = H + , then the corresponding projections are given by pr + = id H + and pr − = 0 which are Fredholm and compact operators, respectively. Therefore we have H + ∈ Gr( ∞ 2 , ∞). Definition 3.2 ([34, Section 5]). We denote by Γ + the set of all continuous functions g : The set Γ + acts on H by pointwise multiplication. In particular, Γ + forms a group. This action induces the action of Γ + on Gr: , satisfying the following: and A : H + → H − is the linear operator such that graph(A) = W .

3.2.
Krichever's construction. In connection with algebraic geometry and infinitedimensional integrable systems, a fundamental observation / construction of Krichever [19,20,21] states that, associated to each non-singular algebraic curve with some additional data (which are called algebro-geometric data) is a solution of the KdV equation. This construction has been developed further, and after a remark by Mumford [30], it was formalised by Segal-Wilson [34] as follows.
In this context, an algebro-geometric datum (X, L, x ∞ , z, ϕ) consists of a complete irreducible complex algebraic curve X, a rank-one torsion-free coherent sheaf L over X, a non-singular point x ∞ ∈ X, a closed neighbourhood X ∞ of x ∞ , a local parameter 1/z : X ∞ → D 0 ⊂ C which sends x ∞ to 0, and ϕ : L| X∞ → D 0 × C is a trivialisation of L| X∞ . Each section of L| X∞ is identified with a complex function on D 0 under ϕ. Let X 0 := X \(the interior of X ∞ ) and then the closed sets X 0 and X ∞ cover X, and X 0 ∩X ∞ is identified with S 1 under z.
Given this algebro-geometric datum, one can associate a closed subspace W ⊂ H consisting of all analytic functions S 1 → C which, under the above identification, extend to a holomorphic section of L on an open neighbourhood of X 0 . More explicitly, one can write s is a holomorphic section on a neighbourhood of X 0 where (1/z) −1 : D 0 → X ∞ is the inverse function of 1/z. It is known that W ∈ Gr (see [34, Proposition 6.1]) and further, in the class where X is a compact Riemann surface (then L is automatically a complex line bundle, hence a maximal torsion-free sheaf), this correspondence (X, L, x ∞ , z, ϕ) → W ∈ Gr is one-to-one (see [34, Proposition 6.2]).
3.3. The appearance of Faber polynomials and Grunsky coefficients. Let f : D 0 → C be a univalent function such that f (0) = 0 and f (D 0 ) is bounded by a Jordan curve. We set β : C → C by β(w) := 1/w. For a subset A ⊂ C, we shall write A −1 := β(A). Let D ∞ := C \ D 0 . We obtain an algebro-geometric datum (X, L, x ∞ , z, ϕ) by setting X = C, L = C × C, x ∞ := ∞, . Here we note that z extends continuously to X ∞ by virtue of the Caratheodory's extension theorem for z.
Therefore we can embed f into the Grassmannian by assigning a Hilbert space W = W f as above. In this case, we have C \ (f (D 0 ) −1 ), and hence In order to start this paper's main calculation, let us specify this more explicitly.
In our case, we put and then we can describe O(X 0 ) by O( D ∞ ) through the transformation As a result, (Ad β * (F)φ)(w) is a power series in 1/w. Actually, in view of the Cauchy integral formula where [(f −1 (w)) −k ] 0 denotes the constant-part plus the principal-part of the Laurent series for (f −1 (w)) −k = (1/f −1 (w)) k . In particular, every element in O( C \ f (D 0 )) can be written as a series of 1 and Q k (w) := [(f −1 (w)) −k ] 0 for k 1. Each Q k is called the k-th Faber polynomial associated to the domain C \ f (D 0 ) (or just simply f ). By definition, Q k (w) is a polynomial of degree n in 1/w.
where z is the identity map on D ∞ . We note that W f = H + in the case of f (z) ≡ z.
Remark 3.1. The Faber polynomials appeared first (with a different formalism, but equivalent to our presentation) in the context of approximations of functions in one complex variable by analytic functions (see [7] and [6]). Since then, they also play an important role in the theory of univalent functions (see [32]). We introduced the Faber polynomials in a slightly non-standard way in order to have them in a form which is suitable for embedding univalent functions into the Grassmannian by using Faber polynomials.
3.4. Action of words. Let X = {x 1 , x 2 , x 3 , · · · } be a countable set of letters. The free monoid X * on X is the set of all words in the letters X, including the empty word ∅. We denote by C X n the free associative and unital C-algebra on X. The unit of this algebra is the empty word which we will denote by 1 := ∅. The set C X n stands for ⊕ w Cw where the summation is taken over all words w of length n.
The action of naturally extends to C X [[z]] and then we call S(ξ(x, z)) s,t := S(ξ(x, z)) s,t the signature of ξ(x, z).
For f ∈ C X ((z −1 )) and x ∈ C X , T (f, x) will be denoted by f. z x in the sequel. The following is clear by definition: Proposition 3.7. T defines an action of the C-algebra C X on C X ((z −1 )) from the right.
The right action T can be extended to the right action under which, the image of (f, z n x ip · · · x i 1 ) is mapped to z n (f. w x ip · · · x i 1 ) =: f. w (z n x ip · · · x i 1 ). Note that now the notation f. w S(x) makes sense.
By tensoring the right action (3.2) it gives rise to under which, the image of (f ⊗ g, x ⊗ y) will be denoted by (f. w x) ⊗ (g. u y) in the sequel. We recall that the tau-function corresponding to W ∈ Gr, is given by e (r+s)x 0 (t) x r+s (w −r . w S(ξ(x, e x 0 (t) ))) ¡ (u −s . u S(ξ(x, e x 0 (t) ))) 0,t .
The proof can be found in Section A.4

Connections with Conformal Field Theory and Free Probability
In this last section, we would like to briefly discuss aspects of the relation the material developed so far has with free probability and conform field theory. Let us first recall D. V. Voiculescu's [38] basic definitions and notions.
A pair (A, ϕ) consisting of an unital algebra A and a linear functional ϕ : A → C with ϕ(1 A ) = 1, is called a non-commutative probability space, and then any element in A is called a random variable.

1.2]).
On the other hand, for any distribution µ ∈ Σ × , is the compositional inverse of the associated moment series of µ: M µ (z) := ∞ n=1 µ(X n )z n . Then for any two distribution µ, ν ∈ Σ × , a new distribution µ ⊠ ν ∈ Σ × can be determined by the relation which is called the multiplicative free convolution of µ and ν ([38, Definition 3.6.1 and Remark 3.6.2]).
If one restricts this result to ν ∈ Σ × 1 , then it follows from results by R. Friedrich and J. McKay [11], that the following diagram commutes: In 2017, R. Friedrich * , pointed out that any distribution of mean one corresponds to a tau-function of the KP hierarchy, extending results obtained with J. McKay [10]. This can be seen as follows: A sequence K = {K n (x 1 , · · · , x n )} n∈N of polynomials K n (x 1 , · · · , x n ) ∈ C[x 1 , · · · , x n ] determines a mapping where Λ(A) := 1 + zC[[z]] and K 0 := 1. The sequence K = {K n (x 1 , · · · , x n )} n∈N is called a (Hirzebruch) multiplicative sequence (or m-sequence over C) if the above mapping is a homomorphism between the monoid Λ(A) (where the product is defined by pointwise multiplication) and the ring of formal power series C[[z]] (but considered as a monoid). A result by T. Katsura, Y. Shimizu and K. Ueno [15] says that, if one defines b n ∈ C as the coefficient of x n in the polynomial K n (x 1 , · · · , x n ), then where t = (t 1 , t 2 , · · · ), is a tau-function of the KP hierarchy ([15, Theorem 5.3, 1)]). Furthermore, the corresponding wave operator belongs to the class 1 + ∂ −1 , all the coefficients are constants, and hence the corresponding Lax operator is given by L = ∂ t 1 ). In particular, in [10,Theorem 4.3] this was related to the infinite Lie group (Σ 1 , ⊠).
Summarising the above, we have the following isomorphisms of commutative groups and their embedding into the universal Grassmannian UGM: (ii) We have for z ∈ D 0 and w ∈ C \ f (D 0 ).
(iii) Q n (w) is a polynomial of degree n in 1/w such that (iv) Q n (w) is a polynomial of degree n in 1/w such that (ii) For each ζ ∈ ∩ 0 t T f t (D), the mapping [0, T ] ∋ t → f −1 t (ζ) ∈ D is continuous and of bounded variation.
Proof. (i) Consider the mapping We will show that, there exists an open neighbourhood D of 0 ∈ C such that ϕ| [0,T ]×D is injective. We divide the proof into two parts.
Definition A.5. Let X and Y be nonempty subsets of C.
We know so far that ϕ| V : V → V is injective and surjective. It remains to prove the continuity of (ϕ| V ) −1 , but this is clear from (A.6) and the fact that h is continuous.
Let γ 1 , γ 2 ⊂ C be two Jordan curves. According to the Jordan curve theorem, we can decompose C \ γ i into a disjoint union i is the unbounded connected component of C \ γ i . By this notation, we have: Lemma A.8. Suppose that both of γ 1 and γ 2 are surrounding the origin, and there exists δ > 0 such that . Proof. Let d H (γ 1 , γ 2 ) =: ε 0 < δ and ε ∈ (ε 0 , δ) be arbitrary. For each subset A ⊂ C, we write A ε := {z ∈ C : d(z, A) < ε}.
Here we shall observe that η 2 ⊂ C ′ 1 does not occur. In fact, suppose that this is the case. Since η 2 ∩ γ 1 = ∅, we have only two cases: C 1 ⊂ (inner domain of η 2 ) or C 1 ∩ (inner domain of η 2 ) = ∅. The first case can not occur since then contradicts to γ 1 ⊂ T ε (γ 2 ). The second case are also impossible because γ 1 and γ 2 surround the origin. Similarly, η 1 ⊂ C ′ 2 does not occur. Therefore, (1) is the only case. It is clear that C ′ 1 ∩ (C ′ 2 ) ε is nonempty and open in C ′ 1 . We shall show that it is also closed in C ′ 1 . Suppose that z n ∈ C ′ 1 ∩ (C ′ 2 ) ε , n ∈ N converges to a point z ∈ C ′ 1 . By (1), we have ∂(C ′ 2 ) ε = η 2 ⊂ C 1 , so that it must be z ∈ (C ′ 2 ) ε .
Henceforth, by taking anew x 1 , x 2 , · · · as non-commutative indeterminates, and b −m,−n 's as polynomials in x i 's, we shall consider the following equation: (roughly speaking, the polynomial b −m,−n means e −(m+n)x 0 (t) b −m,−n (t) and 'applying the indeterminate x k from the right' means 'applying t 0 e −kx 0 (s) dx k (s)× to functions of s') and then we shall make some observations about the equation (A.9) and introduce some notations: If we apply (A.9) to b −m,−n , we get (a) The terms (n−k)b −m,−(n−k) x k and (m−k)b −(m−k),−n x k for each k. We shall denote these phenomena by respectively (Note that the multiplication by the x * 's must sit just right to the next b * , * 's).
(b) The term −x n+m , to which we can not apply (A.9) anymore. This means, consider the situation that we apply (A.9) iteratively to b * , * 's appeared in the previous stage. Suppose we have the term b −m,−n at some stage. Then chasing the term multiplied by −x * which arose from the first term on the right-hand side in (A.9), makes us get out of the loop of iterations. We shall denote this phenomenon by Let k ∈ N be such that 2 k n + m. We shall find the term of the form x k (...) in the polynomial expression of b −m,−n in the x i 's. For this, we shall fix i ∈ {1, · · · , m} and j ∈ {1, · · · , n} such that i + j = k. Suppose that p, q ∈ N and i 1 , · · · , i p , j 1 , · · · , j q ∈ N satisfy i 1 + · · · + i p = m − i and j 1 + · · · + j q = n − j. We then put a r := m − (i 1 + · · · + i r ) for r = 1, · · · , p and c s := n − (j 1 + · · · + j s ) for s = 1, · · · , q. Note that a p = i and c q = j. According to this notation, we can divide the situation into the following three cases: where w i 1 ,··· ,ip;j 1 ,··· ,jq = a 1 a 2 · · · a p = (m − i 1 )(m − (i 1 + i 2 )) · · · (m − (i 1 + i 2 + · · · + i p )).
(3) If there exist such q and (j 1 , · · · , j q ) but not for p and (i 1 , · · · , i p ) (then we have i = m), then the diagram which we can have is the following: Hence we have a single path from b −m,−n to the 'end' in the above diagram. This path produces the term −w j 1 ,··· ,jq x k (x jq · · · x j 2 x j 1 ) where w j 1 ,··· ,jq = c 1 c 2 · · · c q = (n − j 1 )(n − (j 1 + j 2 )) · · · (n − (j 1 + j 2 + · · · + j q )). Now by reinterpreting it in the language of paths x k (t)'s, we conclude the result.