Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 90, pp. 1–32.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.90

NEW APPROACH TO THE LAGRANGE-BÜRMANN THEOREM VIA

OMEGA CALCULUS AND APPLICATIONS

ANTÔNIO FRANCISCO NETO

Abstract. A novel approach to the ubiquitous multidimensional Lagrange-Bürmann Theorem

is developed which uses the omega calculus (OC) developed long ago by MacMahon to study
the partition of natural numbers. Several applications are given including the answer to open

questions regarding the generalized Lambert function W as stated in [56]. More precisely, a

master theorem is presented introducing a new generalized Lambert function for which several
previously known representations arise as special cases including most Taylor series results

of [56] and some other integral representations. Furthermore, the convergence radius of the
aforementioned generalized Lambert function is explicitly determined for which even special

cases were not known before. This work shows another instance where omega calculus is useful,

this time, to address inverse problems of general interest related to functional equations.

1. Introduction

The Lagrange-Bürmann (L-B) formula including its multidimensional version is a distinguished
example of an inverse problem. More precisely, it provides an expansion of the composition of
a function defined by a power series with an inverse function (provided it exists) and has a long
history with many proofs, variations, and widespread applications. We refer the reader to the non-
exhaustive list [1, 11, 26, 27, 28, 29, 31, 37, 40, 44, 48, 55, 60, 67, 69] including the many facets
involving in its proof and extensions to the noncommutative and q-contexts. The applications
are so widespread that even if we limit ourselves to the physical context many scales are involved
ranging from the micro, such as quantum mechanics as in, e.g., [70], to the macro, such as general
relativity as in, e.g., [68].

Another inverse problem comprises the computation of the Lambert function, W (ζ) for short,
which solves the functional equation

W (ζ)eW (ζ) = ζ (1.1)

and extensions [57] sharing the same ubiquitous character of the L-B formula and appearing also
in a wide range of scales from the micro to the macro [14, 57, 16, 61, 65, 66]. In this respect,
for a general account we refer the reader to [57] which includes a detailed analysis of its multi-
valued character and applications in the physical context (see also [59] for a pedagogical account).
Although we focus on the scalar Lambert W function as defined in (1.1) we refer the reader to [72]
for applications of the matrix-valued Lambert W function in order to solve time-delay systems.

Omega Calculus (OC) also known as MacMahon Partition Analysis (MPA) was originally in-
troduced by MacMahon in order to describe the partition of natural numbers [51]. Since then
many extensions [18, 19, 21, 23] and unexpected applications emerge including a new basis and
integral-free approach [20, 24] to the Dyson series [17, 43, 63] associated with the time-ordering op-
erator [25] with dynamics dictated by the Schrödinger equation. In this way, a new combinatorial
approach to perturbation expansion which implies the divided-differences approach of [45, 46] if a
basis is used for the time-independent part of the generator of the dynamics was developed. Aside

2020 Mathematics Subject Classification. 05A15, 05A17, 34K05, 44A99.
Key words and phrases. Lagrange-Bürmann theorem; Omega calculus; Lambert function;

functional equation; inverse problem.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted June 25, 2025. Published September 26, 2025.

1



2 A. F. NETO EJDE-2025/90

from the OC approach to perturbation theory, there are plenty of others combinatorial/algebraic
approaches to perturbation theory each one with distinct flavors [8, 9, 10, 15, 30, 49, 53, 50] and
several relying on graphs [6, 7]. More recently, the inverse problem associated with the dynamics
of non-autonomous systems of differential equations was considered; that is, given the generator of
the dynamics the problem comprises computing perturbatively the associated dynamical system.
More concretely, a version of the aforementioned inverse question was raised as an open problem
in [4] and answered in [22] using OC. In this sense OC provides another operational approach
besides others more well-known such as [43, 54].

In this work, we show that the aforementioned inverse problems; that is, the L-B formula and
the Lambert W function and generalizations are amenable to be treated in the realm of OC with
interesting and unexpected consequences. We will show, among other things, that a generalized
Lambert W function can be constructed which implies several previously known results concerning
representations of the Lambert W functions and associated extensions. This fact justifies our
use of the name master theorem as it encapsulates in a single expression many known results
including Taylor series and integral type representations. Furthermore, the convergence radius
of the aforementioned generalized Lambert W function is explicitly determined for which even
special cases were not known before. The results summarized above answer some of the open
questions put forward in [56, Section 5]. Furthermore, we generalize a previous representation of
a function related to configuration spaces in the context of algebraic geometry [52, Chapter 4] and
a formula for higher order derivatives of the inverse function as described in [42].

This work is organized as follows. In Section 2 the necessary background is introduced and
some central results are stated in Theorems 3.1, 3.5, and 3.7 in Section 3. Some applications are
discussed in Sections 7, 8, 9, and 10. Sections 7 and 8 concern applications of Theorem 3.1, where
we introduce a generalized Lambert W function along with its radius of convergence, respectively,
which implies as special cases several previous cases addressed in the literature including the
answer to some of the open questions raised in [56]. Other applications are discussed in Sections
9 and 10 concerning a sequence arising in algebraic geometry using Theorem 3.5 and higher order
derivatives related to the inverse function using Theorems 3.1 and 3.7, respectively. We finish
with some concluding remarks in Section 11. To make the text more fluid we include the proofs
of Theorems 3.1, 3.5, and 3.7 in Sections 4, 5, and 6, respectively. In doing so, the versatility of
the OC is shown by allowing us to construct multiple proofs of the aforementioned theorems and
auxiliary results with several distinct flavors; that is, exploring different aspects which come along
the OC representation.

2. Auxiliary results

2.1. Notation. Before we continue, we introduce the notation used throughout this work:

• i, j, k, ℓ, m, n, p, q ∈ N.
• a, b, c ∈ Z.
• Greek letters such as α, β, ζ, and so on, stand for complex numbers with the Omega
variables represented by the middle of the Greek alphabet such as λ, µ, and ν. ℜ(α) (ℑ(α))
stands for the real (complex) part of the complex number α so that α = ℜ(α) + ıℑ(α).

• [m,m+ n] = {m, . . . ,m+ n}, [n] = [1, n], and [n]0 = [0, n].
• ϵ, ε = ±1.
• ek stands for the unit vector in Rn with all the coordinates zero except for the k−coordinate

which is one and e =
∑n

k=1 ek.

• F ⟨−1⟩ means the inverse function of F or

F ⟨−1⟩ ◦ F = I = F ◦ F ⟨−1⟩,

where I stands for the identity function and F ⟨n⟩ means function composition or

F ⟨n⟩ = F ◦ · · · ◦ F
with F appearing n times.

• If F (ξ) =
∑

n αnξ
n, then ⟨ξn⟩F (ξ) := αn.

• F (n) ≈ G(n) means asymptotic equality as n → ∞.



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 3

• δa,b stands for the Kronecker delta.
• Sn is the set of permutations of n elements.

2.2. Basics of omega calculus. The OC is rooted in the use of the Omega operators.

Definition 2.1. Let αa ∈ Cn for each a ∈ Zn and λa = λa1
1 . . . λan

n . We define the linear operators
acting on absolutely convergent matrix valued expansions by

λ

Ω
=

∑
αaλ

a := α0n
,

λ

Ω
≥

∑
αaλ

a :=
∑
a≥0

αa

(2.1)

in an open neighbourhood of the complex circles |λi| = 1 with∑
:=

∞∑
a1=−∞

· · ·
∞∑

an=−∞
.

In other words, the Omega operator in (2.1) extracts in a given convergent expansion only
powers of λn such that n = 0. We note that the Omega operators above are connected. Indeed,
e.g., we have

λ

Ω
=
F (λ) =

λ

Ω
≥

(
1− 1

λ

)
F (λ)

which follows from
λ

Ω
=
λa = δa,0 =

λ

Ω
≥

(
1− 1

λ

)
λa.

Definition 2.1 is well-posed in the sense that we can ensure that all expressions considered
here have no singularities in the λi variable in an open neighbourhood of the circle |λi| = 1. As
remarked in [2], this is an important ingredient leading to unique Laurent expansions avoiding
ambiguous results as discussed in [2, Introduction] and in more details below. In Definition 2.1
we can see an example of the elimination procedure which is the basic building block about which
the OC is based. More precisely, the Omega operator in (2.1) selects only the terms in

∑
αaλ

a

with a = 0n resulting in an expression free of Omega variables or, in other words, the Omega
variable λ is eliminated. From now on we refer to the expression containing an Omega variable a
crude generating function. For other aspects of the elimination procedure, we refer the reader to
[3, 33, 71, 23]. Two key observations of the elimination procedure in the context of the OC which
follow directly from Definition 2.1 will be crucial for us here.

The elimination is independent of the order chosen to eliminate the variables; that is, we have

λ

Ω
=

(µ

Ω
=

)
=

λ,µ

Ω
=

=
µ

Ω
=

(λ

Ω
=

)
.

It follows as a consequence that the Omega operator commutes with derivatives of complex vari-
ables not to be eliminated. Indeed, we have the next lemma.

Lemma 2.2. We have

Dα

(λ

Ω
=
F (α, λ)

)
=

λ

Ω
=

(
DαF (α, λ)

)
.

Proof. The proof follows the usual limiting procedure and the linearity of the Omega operator
or by the simple observation that the derivative is the coefficient of the linear term α in F (α, λ)
which can be rewritten as an Omega operator. In symbols, we have

Dα

(λ

Ω
=
F (α, λ)

)
=

µ

Ω
=

1

µ

(λ

Ω
=
F (α+ µ, λ)

)
=

λ

Ω
=

(µ

Ω
=

1

µ
F (α+ µ, λ)

)
=

λ

Ω
=

(
DαF (α, λ)

)
.

□



4 A. F. NETO EJDE-2025/90

Care must be taken about the smaller parameter in a crude generating function containing
Omega variables in order to avoid ambiguity as discussed before. Indeed, on the one hand, observe
that

λ

Ω
=

λ

1− α/λ
= α

if |α| < 1. On the other hand, if |α| > 1, we have

λ

Ω
=

λ

1− α/λ
= − 1

α

λ

Ω
=

λ2

1− λ/α
= 0.

The next lemmas which comprise particular elimination procedures will be useful later on.

Lemma 2.3. We have
λ

Ω
=

λn

(1− αλ)(1− β/λ)
=

βn

1− αβ
.

Proof. Using Definition 2.1 we obtain

λ

Ω
=

λn

(1− αλ)(1− β/λ)
=

λ

Ω
=

(
λn + αλn+1 + α2λn+2 + . . .

)(
1 +

β

λ
+

(β
λ

)2

+ . . .
)

= βn + αβn+1 + αβn+2 + . . .

=
βn

1− αβ
.

□

We also have the important lemma.

Lemma 2.4.

λ

Ω
=

F (αλ)

λn

(
1− β

λ

)−1

=

{
F (n)(0)

n! αn, β = 0
F (αβ)−

∑n−1
m=0 F (m)(0)(αβ)m/m!

βn , β ̸= 0

with the convention
∑−1

m=0 ≡ 0.

Proof. We first show

λ

Ω
=

eαλ

λn

(
1− β

λ

)−1

=

{
αn

n! , β = 0
eαβ−

∑n−1
m=0(αβ)

m/m!
βn , β ̸= 0.

Indeed, note that

λ

Ω
=

eαλ

λn

(
1− β

λ

)−1

=
λ

Ω
=

1

λn

(
1 +

αλ

1!
+

α2λ2

2!
+ . . .

)(
1 +

β

λ
+

β2

λ2
+ . . .

)
=

λ

Ω
=

(
1 +

αλ

1!
+

α2λ2

2!
+ . . .

)( 1

λn
+

β

λn+1
+

β2

λn+2
+ . . .

)
=

αn

n!
+

αn+1β

(n+ 1)!
+

αn+2β2

(n+ 2)!
+ . . .

=

{
αn

n! , β = 0
eαβ−

∑n−1
m=0(αβ)

m/m!
βn , β ̸= 0.

The result now follows using

F (αλ) = lim
k→∞

∑
ℓ≥k

F (ℓ)(0)µℓ exp
(αλ

µ

)
.

□

We also have
λ

Ω
=

F (αλ)

λn

(
1− β

λ

)−m−1

=
Dm

β

m!

λ

Ω
=

F (αλ)

λn−m

(
1− β

λ

)−1

using that we can permute the symbols Ω and Dm
β .



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 5

Lemma 2.5.
λ

Ω
=

F (λ)G(λ)

λk
=

∑
m+n=k

µ,ν

Ω
=

F (µ)

µm

G(ν)

νn
.

Proof. We have

λ

Ω
=

F (λ)G(λ)

λk
=

λ,µ,ν

Ω
=

1

λk

F (µ)

1− λ/µ

G(ν)

1− λ/ν

=
µ,ν

Ω
=

(λ

Ω
=

1

λk

1

1− λ/µ

1

1− λ/ν

)
F (µ)G(ν)

=
∑

m,n≥0

µ,ν

Ω
=

(λ

Ω
=
λm+n−k

)
︸ ︷︷ ︸

=δm+n,k

F (µ)

µm

G(ν)

νn

=
∑

m+n=k

µ,ν

Ω
=

F (µ)

µm

G(ν)

νn
,

where the first equality follows using Lemma 2.4 with n = 0. □

Finally, we recall the connection between OC and standard integral representations which
follows from a direct observation,∫

eımθdθ = 2πδm,0 = 2π
λ

Ω
=
λm = 2π

λ

Ω
≥

(
1− 1

λ

)
λm,

where
∫
=

∫ π

−π
,
∫ 2π

0
. In this way, we can write∫

f(cos θ, sin θ)dθ =

∫
g
(
eıθ, e−ıθ

)
dθ = 2π

λ

Ω
=
g
(
λ, λ−1

)
= 2π

λ

Ω
≥

(
1− 1

λ

)
g
(
λ, λ−1

)
(2.2)

provided g satisfies the conditions stated in Definition 2.1. This observation was used in [23] to
solve a nontrivial integral using OC.

Example 2.6 ([64, Problem 2202]). We have

I+ =

∫ 2π

0

cos(cos(θ)) cosh(sin(θ))dθ = 2π,

I− =

∫ 2π

0

sin(cos(θ)) cosh(sin(θ))dθ = 0.

Indeed, if λϵ := ı(λ+ ϵλ−1)/2, we note that

Iϵ = 2π
λ

Ω
=

eλ+ + ϵe−λ+

2

eλ− + e−λ−

2

=
π

2

λ

Ω
=

(
eıλ + eıλ

−1

+ ϵe−ıλ + ϵe−ıλ−1
)
= 2πδϵ,+.

Example 2.7. We have

I =

∫ π

0

cos(θ)

1 + α cos(θ)
dθ = π

√
1− α2 − 1

α
√
1− α2

.

Indeed, we note that

I =
1

2

∫ π

−π

cos(θ)

1 + α cos(θ)
dθ =

π

2

λ

Ω
=

λ+ λ−1

1 + α(λ+ λ−1)/2
.

Next, we introduce β2
+ + β2

− = 1, β+β− = −α/2, and assume |β−| < |β+|. A simple calculation
gives

βϵ =
α+ + ϵα−

2



6 A. F. NETO EJDE-2025/90

such that αϵ :=
√
1 + ϵα2. In this way, we have

I =
π

2

λ

Ω
=

λ+ λ−1

(β+ − β−λ)(β+ − β−λ−1)

= π
λ

Ω
=

λ

(β+ − β−λ)(β+ − β−λ−1)

=
πβ−

β2
+ − β2

−

= π

√
1− α2 − 1

α
√
1− α2

,

where the second equality follows from the invariance of the denominator under the replacement
λ → λ−1 and, the last but one follows from Lemma 2.3. Although this example is of elementary
nature it is sufficient to show another nice feature of OC; that is, the ability to obtain closed form
expressions.

3. Lagrange-Bürmann expansion via OC

In all the expansions that follow care must be taken such that Definition 2.1 applies. For this
purpose we take

|ζ| < |λ|.
We let

F (ζ) =
∑
n≥0

αnζ
n, (3.1)

G(ζ) =
∑
n≥1

βnζ
n (3.2)

with β1 ̸= 0. We can now state the main results of this section.

Theorem 3.1. We have(
F ◦G⟨−1⟩)(ζ) = α0 −

λ

Ω
=
λF ′(λ) ln

(
1− ζ

G(λ)

)
. (3.3)

If we set F = I in Theorem 3.1 we immediately obtain the Lagrange inversion formula stated
in the next corollary.

Corollary 3.2. We have

G⟨−1⟩(ζ) = −
λ

Ω
=
λ ln

(
1− ζ

G(λ)

)
.

To get a flavor of what is going on we describe some simple examples before embarking into
more involved calculations.

Example 3.3. We take

G(ζ) = ζ exp(ζ) (3.4)

to obtain

G⟨−1⟩(ζ) = −
λ

Ω
=
λ ln

(
1− ζ

λ exp(λ)

)
=

∑
n≥1

ζn

n

λ

Ω
=

exp(−nλ)

λn−1
=

∑
n≥1

(−n)n−1

n!
ζn

with radius of convergence

R = lim
n→∞

∣∣ γn
γn+1

∣∣ = 1

e
,

where γn = (−n)n−1/n!. From now on we write G⟨−1⟩(ζ) = W (ζ) for G as in (3.4) using the
standard notation for the Lambert W function.



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 7

Example 3.4. We take

G(ζ) =
ζ

(1− ζ)1/3

to obtain

G⟨−1⟩(ζ) = −
λ

Ω
=
λ ln

(
1− ζ(1− λ)1/3

λ

)
=

∑
n≥1

ζn

n

λ

Ω
=

(1− λ)n/3

λn−1
=

∑
n≥1

(−1)n−1

n

(
n/3

n− 1

)
ζn

with radius of convergence

R = lim
n→∞

∣∣ γn
γn+1

∣∣ = 3

e
,

where

γn =
(−1)n−1

n

(
n/3

n− 1

)
=

(−1)n−1

n

λ

Ω
=

(1− λ)n/3

λn−1

=
(−1)n−1nn−1

n

λ

Ω
=

(1− λ/n)n/3

λn−1

≈ (−1)n−1nn−1

n

λ

Ω
=

exp(−λ/3)

λn−1

≈ nn−1

n!3n−1

using rescaling λ → λ/n to obtain the third equality and observing(
1 +

α

n

)n

≈ eα. (3.5)

Alternatively, we can state the following Omega representation which avoids the direct compu-
tation of F ′.

Theorem 3.5. We have (
F ◦G⟨−1⟩)(ζ) = λ

Ω
=

λF (λ)G′(λ)(
1− ζ/G(λ)

)
G(λ)

. (3.6)

Again, if we set F = I in Theorem 3.5 we immediately obtain the next corollary.

Corollary 3.6. We have

G⟨−1⟩(ζ) =
λ

Ω
=

λ2G′(λ)(
1− ζ/G(λ)

)
G(λ)

.

We now turn to a multivariable extension of Theorem 3.5. We take

|ζk∈[n]| < |λk∈[n]|,
Gi(ζ) = βiζi(1 +Hi(ζ)),

where ζ = (ζ1, . . . , ζn) and Hi(ζ) = O(ζk) with |k| ≥ 1 such that we write

Hi(ζ) =
∑
j

βijζ
j.

Theorem 3.7. We have(
F ◦G⟨−1⟩)(ζ) = λ

Ω
=

λeF (λ) det
(
DiGj(λ)

)∏n
k=1

(
1− ζk/Gk(λ)

)
Gk(λ)

. (3.7)



8 A. F. NETO EJDE-2025/90

Of course Theorem 3.5 follows from Theorem 3.7, but we prefer to state them separately for
readability because the proof of Theorem 3.7 given in Section 6 uses heavier notation and it is
motivated by the proof of Theorem 3.5 in Section 5.

We now establish contact with [34, Theorem 1] by observing that if

F (λ) =
∑
a∈Z

αaλ
a

is as in Definition 2.1, then

α−1 =: Res(F ) =
λ

Ω
=
λF (λ).

Therefore, using Theorem 3.1 we have(
F ◦G⟨−1⟩)(ζ) = α0 +

∑
n≥1

1

n
Res

( F ′

Gn

)
ζn

which is nothing more than [34, Theorem 1] in disguise; that is, once the correspondence between
notations is established the equivalence is clear. A similar consideration applies to the multidi-
mensional version in Theorem 3.7; that is, Theorem 3.7 is equivalent to [35, Theorem 7]. More
precisely, Theorem 3.7 is equivalent to

(
F ◦G⟨−1⟩)(ζ) = ∑

n≥0

Res
(F det

(
DiGj

)
Gn+e

)
ζn,

where we now identify

α−e =: Res(F ) =
λ

Ω
=
λeF (λ)

with

F (λ) =
∑
a∈Zn

αaλ
a.

See also [36].
If G(ζ) = ζ/H(ζ) (a functional dependence compatible with (3.2)) with H(0) ̸= 0, then we can

use Theorem 3.1 to obtain well-known equivalent forms of the L-B theorem. Indeed, we have

⟨ζn⟩
(
F ◦G⟨−1⟩)(ζ) = 1

n

λ

Ω
=

λF ′(λ)

(G(λ))n
=

1

n

λ

Ω
=

F ′(λ)(H(λ))n

λn−1
=

1

n
⟨ζn−1⟩F ′(ζ)(H(ζ))n

in agreement with [29, Equation (2.1.1)]. If instead Theorem 3.5 is used, we obtain

⟨ζn⟩
(
F ◦G⟨−1⟩)(ζ) = λ

Ω
=

λF (λ)G′(λ)

(G(λ))n+1

=
λ

Ω
=

F (λ)(H(λ)− λH ′(λ))(H(λ))n−1

λn

= ⟨ζn⟩F (ζ)(H(ζ)− ζH ′(ζ))(H(ζ))n−1

in agreement with [29, Equation (2.1.2)].

4. Proof of Theorem 3.1

We first introduce the key auxiliary result

λ

Ω
=

λG′(λ)

(G(λ))m+1
= δm,0. (4.1)



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 9

First Proof of (4.1). Indeed, if m = 0, then

λ

Ω
=

λG′(λ)

(G(λ))m+1
=

λ

Ω
=

λG′(λ)

G(λ)
= 1.

If m > 0, then

λ

Ω
=

λG′(λ)

(G(λ))m+1
= − 1

m

λ

Ω
=
λD(G(λ))−m = − 1

m

λ,µ

Ω
=

λ

µ(G(λ+ µ))m

with |µ| < |λ| since
λ,µ

Ω
=

λ

µ
(λ+ µ)a = 0.

Indeed, if a ≥ 0 the result is direct since there is no term independent of λ and µ. If a = −n < 0
we write

λ,µ

Ω
=

λ

µ(λ+ µ)n
=

λ,µ

Ω
=

1

µλn−1(1 + µ/λ)n
=

∑
m≥0

(
m+ n− 1

n− 1

)
λ,µ

Ω
=

µm−1

λm+n−1
= 0

since we cannot have m− 1 = 0 = m+ n− 1, because n > 0.

Second proof of (4.1). An even more compact approach is possible showing another instance
where the power of the OC based approach is manifested. For m > 0 and α ∈ C \ {0}, we have

λ

Ω
=

λG′(λ)

α(G(λ))m+1
=

λ

Ω
=

λG′(αλ)

(G(αλ))m+1

= − 1

m

λ

Ω
=
Dα(G(αλ))−m

= −Dα

m

λ

Ω
=
(G(αλ))−m

= −Dα

m

λ

Ω
=
(G(λ))−m = 0

using invariance of the crude generating function under rescaling λ → αλ to obtain the first
equality and observing that (G(λ))−m does not depend on α to obtain the last one. Since α ̸= 0
we obtain the desired result.

Third proof of (4.1). Another compact proof is available using basic OC properties. For m > 0,
we write

G′(λ) = lim
δ→0

G((1 + δ)λ)−G(λ)

δλ
to obtain

λ

Ω
=

λG′(λ)

(G(λ))m+1
= lim

δ→0

λ

Ω
=

G((1 + δ)λ)−G(λ)

δ(G(λ))m+1

= lim
δ→0

(λ

Ω
=

G((1 + δ)λ)

δ(G(λ))m+1
−

λ

Ω
=

G(λ)

δ(G(λ))m+1

)
= lim

δ→0

(λ

Ω
=

G((1 + δ)λ)

δ(G(λ))m+1
−

λ

Ω
=

G((1 + δ)λ)

δ(G((1 + δ)λ))m+1

)
= lim

δ→0

(λ

Ω
=

G((1 + δ)λ)

δ(G(λ))m+1
−

λ

Ω
=

G((1 + δ)λ)

δ(G(λ) + δλG′(λ))m+1

)
= lim

δ→0

(λ

Ω
=

G((1 + δ)λ)

δ(G(λ))m+1
−

λ

Ω
=

G((1 + δ)λ)

δ(G(λ))m+1(1 + δλG′(λ)/G(λ))m+1

)
,

where the third equality follows from invariance under the rescaling λ → (1 + δ)λ. Next, we can
write (

1 +
δλG′(λ)

G(λ)

)−m−1

= 1− (m+ 1)
δλG′(λ)

G(λ)
+O

(
δ2
)



10 A. F. NETO EJDE-2025/90

taking into account only terms linear in δ. Therefore, we have

λ

Ω
=

λG′(λ)

(G(λ))m+1
= (m+ 1) lim

δ→0

λ

Ω
=

λG((1 + δ)λ)G′(λ)

(G(λ))m+2
= (m+ 1)

λ

Ω
=

λG′(λ)

(G(λ))m+1

which implies (4.1) (recall that m > 0).

First proof of Theorem 3.1. We prove Theorem 3.1 by showing that(
Hk ◦

(
G ◦ F ⟨−1⟩

k

))
(ζ) = ζ =

((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ)

or, equivalently, that

Hk ◦
(
G ◦ F ⟨−1⟩

k

)
= I =

(
G ◦ F ⟨−1⟩

k

)
◦Hk,

where I is the identity operator and Hk given by the RHS of (3.3) with Fk(ζ) = ζk so that

F
⟨−1⟩
k (ζ) = ζ1/k. We first show that Hk is a left inverse of G ◦ F ⟨−1⟩

k . We will show an equivalent
statement (

Hk ◦
(
G ◦ F ⟨−1⟩

k

))′
(ζ) = 1,

(
Hk ◦

(
G ◦ F ⟨−1⟩

k

))
(0) = 1.

Therefore, we conclude that

Hk =
(
G ◦ F ⟨−1⟩

k

)⟨−1⟩
= Fk ◦G⟨−1⟩

using the uniqueness of the inverse function. We first observe that(
Hk ◦

(
G ◦ F ⟨−1⟩

k

))
(ζ) = Hk

(
G
(
ζ1/k

))
= −k

λ

Ω
=
λk ln

(
1−

G
(
ζ1/k

)
G(λ)

)
by recalling that α0 = 0 in this case. Indeed, we have(

Hk ◦
(
G ◦ F ⟨−1⟩

k

))′
(ζ) = H ′

k

(
G
(
ζ1/k

))
G′(ζ1/k)(ζ1/k)′

=
λ

Ω
=

kλkG′(ζ1/k)(ζ1/k)′
G(λ)−G

(
ζ1/k

)
= ζ(1−k)/k

λ

Ω
=

λkG′(ζ1/k)
G(λ)−G

(
ζ1/k

)
= ζ(1−k)/k

λ

Ω
=

λkG′(ζ1/k)(
λ− ζ1/k

)(
G(λ)−G

(
ζ1/k

))
/
(
λ− ζ1/k

)
= ζ(1−k)/k

λ

Ω
=

λk−1G′(ζ1/k)(
1− ζ1/k/λ

)(
G(λ)−G

(
ζ1/k

))
/
(
λ− ζ1/k

)
= ζ(1−k)/kζ(k−1)/k︸ ︷︷ ︸

=1

G′(ζ1/k)
limλ→ζ1/k

(
G(λ)−G

(
ζ1/k

))
/
(
λ− ζ1/k

)︸ ︷︷ ︸
=1

= 1.

Now we show that Hk is a right inverse of G ◦ F ⟨−1⟩
k . We have((

G ◦ F ⟨−1⟩
k

)
◦Hk

)
(0) =

(
G ◦ F ⟨−1⟩

k

)(
− k

λ

Ω
=
λk ln(1)

)
= G(F

⟨−1⟩
k (0)) = G(0) = 0

which implies ((
G ◦ F ⟨−1⟩

k

)
◦Hk)(ζ) = O(ζ). (4.2)

Next, observe that

Hk = I ◦Hk =
(
Hk ◦

(
G ◦ F ⟨−1⟩

k

))︸ ︷︷ ︸
=I

◦Hk = Hk ◦
((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(4.3)



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 11

using that Hk is a left inverse of G ◦ F ⟨−1⟩
k which implies

λ

Ω
=
λk ln

(
1− ζ

G(λ)

)
︸ ︷︷ ︸

=−Hk(ζ)

=
λ

Ω
=
λk ln

(
1−

((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ)

G(λ)

)
︸ ︷︷ ︸

=−
(
Hk◦

((
G◦F ⟨−1⟩

k

)
◦Hk

))
(ζ)

. (4.4)

By using (4.2) we can equate terms with equal powers of ζ in (4.4) to obtain the first non-null
contribution

ζk
λ

Ω
=

λk

(G(λ))k
=

k∑
n=1

(((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ)

)n λ

Ω
=

λk

(G(λ))n
.

Next, we use

λ

Ω
=

λk

(G(λ))n
=

λ

Ω
=

λk−n

βn
1 (G(λ)/λ)n

=
λ

Ω
=

λk−n

βn
1

(
1 + β2λ/β1 + β3λ2/β1 + . . .

)n = 0

if n < k and observe that

λ

Ω
=

λk

(G(λ))k
=

1

βk
1

̸= 0

if n = k. We arrive at

ζk
λ

Ω
=

λk

(G(λ))k
=

(((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ)

)k λ

Ω
=

λk

(G(λ))k

=⇒
((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ) = ζ.

Therefore, Hk = Fk ◦G⟨−1⟩ using the uniqueness of the inverse function. Finally, the result now
follows by observing that F in (3.1) is a linear combination of the Fk’s. Since Fk ◦ G⟨−1⟩, the
general case follows straightforwardly recalling that(

F ◦G⟨−1⟩)(ζ) = α0 +
∑
k≥0

αk

(
Fk ◦G⟨−1⟩)(ζ)

and using (3.1) with F =
∑

k≥0 αkFk. □

Second proof of Theorem 3.1. We first observe that(
F ◦G⟨−1⟩)(ζ) = ∑

n≥0

γnζ
n

so that

F (ζ) =
∑
n≥0

γn(G(ζ))n =⇒ F ′(ζ) =
∑
n≥1

nγn(G(ζ))n−1G′(ζ)

to obtain

F ′(ζ)

(G(ζ))m+1
=

∑
n≥0

nγn
G′(ζ)

(G(ζ))m−n+1
.

Using the linearity of the Omega operator along with (4.1) we obtain

λ

Ω
=

λF ′(λ)

(G(λ))m+1
=

∑
n≥0

nγn
λ

Ω
=

λG′(λ)

(G(λ))m−n+1
= mγm

which is equivalent to the required result. □



12 A. F. NETO EJDE-2025/90

5. Proof of Theorem 3.5

First proof of Theorem 3.5. We prove Theorem 3.5 using the same strategy of the previous
section. More precisely, we show that(

Hk ◦
(
G ◦ F ⟨−1⟩

k

))
(ζ) = ζ =

((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ),

where Hk now stands for the right-hand side of (3.6) and Fk(ζ) = ζk so that F
⟨−1⟩
k (ζ) = ζ1/k. We

first show that (
Hk ◦

(
G ◦ F ⟨−1⟩

k

))
(ζ) = ζ.

Indeed, we have(
Hk ◦

(
G ◦ F ⟨−1⟩

k

))
(ζ) = Hk

(
G
(
ζ1/k

))
=

λ

Ω
=

λFk(λ)G
′(λ)(

1−G
(
ζ1/k

)
/G(λ)

)
G(λ)

=
λ

Ω
=

λk+1G′(λ)(
1−G

(
ζ1/k

)
/G(λ)

)
G(λ)

=
λ

Ω
=

λk+1G′(λ)(
λ− ζ1/k

)(
G(λ)−G

(
ζ1/k

))
/
(
λ− ζ1/k

)
=

λ

Ω
=

λkG′(λ)(
1− ζ1/k/λ

)(
G(λ)−G

(
ζ1/k

))
/
(
λ− ζ1/k

)
= lim

λ→ζ1/k

λkG′(λ)(
G(λ)−G

(
ζ1/k

))
/
(
λ− ζ1/k

)
= ζ.

Next, we observe that ((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(0) = G

(
F

⟨−1⟩
k (0)

)
= G(0) = 0

which implies ((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ) = O(ζ)

and use (4.3) to obtain

λ

Ω
=

λk+1G′(λ)(
1− ζ/G(λ)

)
G(λ)

=
λ

Ω
=

λk+1G′(λ)(
1−

((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ)/G(λ)

)
G(λ)

.

Expanding in ζ we obtain the first non-null contribution

ζk
λ

Ω
=

λk+1G′(λ)

(G(λ))k+1
=

(((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ)

)k λ

Ω
=

λk+1G′(λ)

(G(λ))k+1

since

λ

Ω
=

λk+1G′(λ)

(G(λ))k+1
=

1

βk
1

̸= 0.

It follows that ((
G ◦ F ⟨−1⟩

k

)
◦Hk

)
(ζ) = ζ

and the proof is complete, again, after recalling the uniqueness of the inverse and observing that
F in (3.1) is a linear combination of the Fk’s. □



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 13

Second proof of Theorem 3.5. We first observe that(
F ◦G⟨−1⟩)(ζ) = ∑

n≥0

γnζ
n

so that

F (ζ) =
∑
n≥0

γn(G(ζ))n

to obtain
F (ζ)G′(ζ)

(G(ζ))m
=

∑
n≥0

γnG
′(ζ)

(G(ζ))m−n+1
.

Using the linearity of the Omega operator and (4.1) we obtain

λ

Ω
=

λF (λ)G′(λ)

(G(λ))m
=

∑
n≥0

γn
λ

Ω
=

λG′(λ)

(G(λ))m−n+1
= γm.

□

6. Proof of Theorem 3.7

The proof follows along the lines of the proof of Theorem 3.5 in Section 5, but somewhat more
involved. Again, we first introduce the key auxiliary result that follows which is the multidimen-
sional counterpart of (4.1)

λ

Ω
=

λe det
(
DiGj(λ)

)
(G(λ))m+e

= δm,0. (6.1)

First Proof of (6.1). The case m = 0 is handled as follows

λ

Ω
=

λe det
(
DiGj(λ)

)
(G(λ))e

=
λ

Ω
=

det
(
βiδij(1 +Hi(λ)) + βjλjDiHj(λ)

)∏n
k=1 βk(1 +Hk(λ))

=
det(βiδij)∏n

k=1 βk
= 1

using

DiGj(λ) = Di

(
βjλj(1 +Hj(λ))

)
= βiδij(1 +Hi(λ)) + βjλjDiHj(λ)

and observing that in the numerator and denominator only positive powers of λi contribute due
to the functional dependence of Hi(λ).

We next consider the case m > 0 since if some mi = 0 then we have a trivial cancelation of the
omega variable λi resulting in an expression free of the Omega variable λi. We have

λ

Ω
=

λe det
(
DiGj(λ)

)
(G(λ))m+e

=
λ

Ω
=

λe ∑
π∈Sn

sign(π)
∏n

i=1 DiGπ(i)(λ)

(G(λ))m+e

=
(−1)n∏n
i=1 mi

λ

Ω
=
λe

∑
π∈Sn

sign(π)

n∏
i=1

DiG
−mπ(i)

π(i) (λ)

=
(−1)n∏n
i=1 mi

λ,µ

Ω
=

λe

µe

∑
π∈Sn

sign(π)

n∏
i=1

G
−mπ(i)

π(i) (λ+ µiei) = 0

if m > 0. Note that each term of∑
π∈Sn

sign(π)
λ,µ

Ω
=

λe

µe

n∏
i=1

G
−mπ(i)

π(i) (λ+ µiei)

is of the type∑
π∈Sn

sign(π)
λ,µ

Ω
=

λe

µe

n∏
i=1

(λ+ µiei)
aπ(i) =

∑
π∈Sn

sign(π)
λ,µ

Ω
=

λe

µe

n∏
i=1

λ
∑

j aij

i

(
1 +

µi

λi

)aiπ(i)

=
∑
π∈Sn

sign(π)
λ

Ω
=
λe

n∏
i=1

λ
∑

j aij

i

aiπ(i)

λi



14 A. F. NETO EJDE-2025/90

=
∑
π∈Sn

sign(π)aiπ(i)
λ

Ω
=

n∏
i=1

λ
∑

j aij

i = 0

using

(λ+ µiei)
aπ(i) = λ

a1π(i)

1 . . . (λi + µi)
aiπ(i) . . . λ

anπ(i)
n

= λ
a1π(i)

1 . . . λ
aiπ(i)

i

(
1 +

µi

λi

)aiπ(i)

. . . λ
anπ(i)
n

to obtain
n∏

i=1

(λ+ µiei)
aπ(i) = λ

∑
j a1π(j)

1 . . . λ
∑

j aiπ(j)

i

n∏
i=1

(
1 +

µi

λi

)aiπ(i)

. . . λ
∑

j anπ(j)

n

=

n∏
i=1

λ
∑

j aij

i

(
1 +

µi

λi

)aiπ(i)

and
µi

Ω
=

1

µi

(
1 +

µi

λi

)aiπ(i)

=
aiπ(i)

λi
,

where aπ(i) is the π(i) ∈ [n] column of

A := (a1, . . . ,an).

Finally, since the λ variable forces a linear dependence among the rows of A, we obtain a zero
determinant.

Second Proof of (6.1). Another compact proof is available using basic OC properties following
the proof of (4.1). For m1 > 0, we write

DkG1(λ) = lim
δk→0

G1(λ+ δkλkek)−G1(λ)

δkλk

and by Laplace expanding det(DiGj(λ)) along the first column we get

λ

Ω
=

λe det
(
DiGj(λ)

)
(G(λ))m+e

=

n∑
k=1

(−1)k+1
λ

Ω
=

λkDkG1(λ)Fk(λ)

(G1(λ))m1+1
,

where

Fk(λ) :=
λe−ek det

(
DiGj(λ)

)
(G(λ))m+e−(m1+1)e1

∣∣∣∣
(i,j)̸=(k,1)

.

In this way, we obtain

λ

Ω
=

λkDkG1(λ)Fk(λ)

(G1(λ))m1+1

= lim
δk→0

(λ

Ω
=

G1(λ+ δkλkek)Fk(λ)

δk(G1(λ))m1+1
−

λ

Ω
=

G1(λ)Fk(λ)

δk(G1(λ))m1+1

)
= lim

δk→0

(λ

Ω
=

G1(λ+ δkλkek)Fk(λ)

δk(G1(λ))m1+1
−

λ

Ω
=

G1(λ+ δkλkek)Fk(λ+ δkλkek)

δk(G1(λ+ δkλkek))m1+1

)
.

Next, we can write

X(λ+ δkλkek) = X(λ) + δkλkDkX(λ) +O
(
δ2k
)

with X = Fk, G1 and(
1 +

δkλkDkG1(λ)

G1(λ)

)−m1−1

= 1− (m1 + 1)
δkλkDkG1(λ)

G1(λ)
+O

(
δ2k
)

taking into account only terms linear in δ. Therefore, we have
n∑

k=1

(−1)k+1Dk

λi∈[n]\{k}

Ω
=

(λe−ek det
(
DiGj(λ)

)
(G(λ))m+e−(m1+1)e1

)∣∣∣∣
(i,j)̸=(k,1)

= 0.



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 15

Indeed, we use induction with the base case given by (4.1) and the induction hypothesis given by

λi∈[n]\{1}

Ω
=

(λe−e1 det
(
DiGj(λ)

)
(G(λ))m+e−(m1+1)e1

)∣∣∣
(i,j)̸=(1,1)

= δm−m1e1,0

along with a simple observation

λi∈[n]\{k}

Ω
=

(λe−ek det
(
DiGj(λ)

)
(G(λ))m+e−(m1+1)e1

)∣∣∣∣
(i,j) ̸=(k,1)

= 0

for k ̸= 1 because we will always have a prefactor λ1 in λe−ek and no term containing λ−1
1 . Finally,

we obtain

λ

Ω
=

λe det
(
DiGj(λ)

)
(G(λ))m+e

= (m1 + 1)
λ

Ω
=

λe det
(
DiGj(λ)

)
(G(λ))m+e

which implies (6.1) (recall that m1 > 0).

Proof of Theorem 3.7. We can now prove Theorem 3.7. We have(
F ◦G⟨−1⟩)(ζ) = ∑

n∈Nn
0

γnζ
n

so that

F (ζ) =
∑
n∈Nn

0

γn(G(ζ))n

to obtain

F (ζ) det
(
DiGj(ζ)

)
(G(ζ))m+e

=
∑
n∈Nn

0

γn det
(
DiGj(ζ)

)
(G(ζ))m−n+e

.

Using the linearity of the Omega operator along with (6.1) we obtain

λ

Ω
=

λeF (λ) det
(
DiGj(λ)

)
(G(λ))m+e

=
∑
n∈Nn

0

γn
λ

Ω
=

λe det
(
DiGj(λ)

)
(G(λ))m−n+e

= γm

which implies the desired result. □

Remark 6.1. It is instructive to compare our proof with the one given in [35, Theorem 7]. We
note that our proof is much simpler and direct. In particular, the handling of the case ki = 0 is
simpler in the context of our approach as one may compare the proof of [35, Theorem 7] with the
proof above.

7. A new generalized Lambert W function

In this section we introduce a new generalized Lambert W function, and we show that some of
the main results of [56, 58, 12] follows from our new representation. This section is motivated by
the following question left open in [56] which is quoted verbatim below:

Q1: Could one carry out some general analysis for W
(

τ1...τr
σ1...σs

; ζ
)
?

An even more recent reference [57] refers to Q1 as “elusive”.

7.1. Main theorem. The aforementioned function W
(

τ1...τr
σ1...σs

; ζ
)
solves the equation

eξ
∏r

p=1(ξ − τp)∏s
q=1(ξ − σq)

= ζ (7.1)

in the variable ξ and we have

ξ = W
( τ1 . . . τr
σ1 . . . σs

; ζ
)

(7.2)



16 A. F. NETO EJDE-2025/90

using the notation of [57]. Equation (7.1) arise, e.g., in the context of Bose-Femi mixtures [56]
and delay differential equations [59]. More generally, we will consider

exp
( d∑

i=2

αi(ξ − τ1)
i
)
(exp(β(ξ − τ1)) + ρ)eγξ

∏r
p=1(ξ − τp)∏s
q=1(ξ − σq)

= ζ

and we are interested in determining ξ that solves the equation above in the variable ζ. From now
on regarding such ξ we use the notation

ξ = Wρ

( τ
σ
;α, β, γ, ζ

)
,

where τ = (τp)p∈[r] and σ = (σq)q∈[s]. If

G(ξ) = exp
( d∑

i=2

αiξ
i
)
(exp(βξ) + ρ) exp(γξ + δ)

ξ
∏r

p=2(ξ + Tp)∏s
q=1(ξ + Sq)

,

where δ := γτ1, Tp := τ1 − τp, and Sq := τ1 − σq. We obtain

Wρ

( τ
σ
;α, β, γ, ζ

)
= τ1 +G⟨−1⟩(ζ) (7.3)

with the convention

Wρ

(−
σ
;α, β, γ, ζ

)
in (7.3) setting

r∏
p=2

(ξ + Tp) → 1.

Likewise, we use

Wρ

( τ

−
;α, β, γ, ζ

)
meaning (7.3) upon setting

s∏
q=1

(ξ + Sq) → 1.

The explicit determination of G⟨−1⟩(ζ) in (7.3) is the content of the next theorem. We highlight

that it is possible to obtain an explicit expression for Wρ

(
τ
σ ;α, β, γ, ζ

)
in a way that several

theorems emerge as special cases including

W0

(−
−
;0, 0, 1, ζ

)
= W (ζ)

as in Example 3.3,

W0

( τ
σ
;0, 0, 1, ζ

)
= W

( τ1 . . . τr
σ1 . . . σs

; ζ
)

as in (7.2), and

Wρ

(−
−
;0, 0, 1, ζ

)
= Wρ(ζ)

for the ρ-Lambert W function.
To introduce our next result we recall the definition of the Stirling numbers of the second kind

(see, e.g., [13, Chapter 5, Sections 1 and 2]). The Stirling numbers of the second kind
{

n
m

}
count

the number of ways a set of n elements can be partitioned into m nonempty disjoint subsets. They
can be described via an exponential generating function∑

n≥m

{ n

m

}ξn

n!
=

(eξ − 1)m

m!
(7.4)

and adopting the convention { n

m

}
= 0



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 17

if n < m. We will also need later on the rising factorial

ξn := ξ(ξ + 1) . . . (ξ + n− 1) (7.5)

with the convention ξ0 = 1 and (
ℓ

ℓ

)
:=

ℓ!∏d
i=1 ℓi!

(7.6)

stands for the multinomial coefficient.
We are now ready to state the main result of this section giving our answer to Q1. In this

section from now on and the next section, we assume i ∈ [2, d], p ∈ [2, r], and q ∈ [s] unless
otherwise stated.

Theorem 7.1. We have the Omega representations

Wρ

( τ
σ
;α, β, γ, ζ

)
= τ1 −

λ

Ω
=
λ ln

(
1− ζ

G(λ)

)
, (7.7)

Wρ

( τ
σ
;α, β, γ, ζ

)
=

λ

Ω
=

λ2G′(λ)(
1− ζ/G(λ)

)
G(λ)

, (7.8)

where

G−1(λ) =

∏
q(λ+ Sq)

λ exp
(∑

i αiλi + γλ+ δ
)
(eβλ + ρ)

∏
p(λ+ Tp)

.

Equivalently, if ρ ̸= −1, we have the Omega free Taylor series expansion

Wρ

( τ
σ
;α, β, γ, ζ

)
= τ1 +

∑ (
ζe−δ

)n
n

(−n)m

m!

(
m

ℓ

)(
ℓ

ℓ

)(
k + n− 1

n− 1

)
× (−1)k+

∑
p mp

{ j

k

}k!

j!

αℓβjγm−ℓ

(1 + ρ)k+n
a(n)n b(n)m

Sn(∏
p T

n
p

)
Tm

,

(7.9)

where

j := n−m+ ℓ−
∑
i

iℓi − sn− 1−
∑
p

mp +
∑
q

nq,

∑
:=

∑
n≥1

∑
m≥0

m∑
ℓ=0

∑
ℓ

j∑
k=0

∑
m,n≥0

with ∑
ℓ

:=
∑

ℓ2+···+ℓd=ℓ

, a(n)n :=
∏
q

(
n

nq

)
with n = (nq) such that Sn =

∏
q S

nq
q , and

b(n)m :=
∏
p

(
mp + n− 1

n− 1

)
with m = (mp) such that Tm =

∏
p T

mp
p .

Proof. We observe that (7.7) and (7.8) follow directly from Theorems 3.1 and 3.5 applied to (7.3),
respectively.

We now turn to the proof of (7.9). Starting with (7.7) we have

Wρ

( τ
σ
;α, β, γ, ζ

)
= −

λ

Ω
=
λ ln

(
1−

ζ
∏

q(λ+ Sq)

λ exp
(∑

i αiλi + γλ+ δ
)
(eβλ + ρ)

∏
p(λ+ Tp)

)

=
∑
n≥1

ζn

n
exp(−nδ)

λ

Ω
=

exp
(
− n

(∑
i αiλ

i + γλ
))

λn−1(eβλ + ρ)n

∏
q(λ+ Sq)

n∏
p(λ+ Tp)n



18 A. F. NETO EJDE-2025/90

using

(λ+ Sq)
n =

n∑
nq=0

(
n

nq

)
λn−nqSnq

q ,

we write
1

(λ+ Tp)n
=

1

Tn
p (1 + λ/Tp)n

such that
1

(1 + λ/Tp)n
=

∑
mp≥0

(
mp + n− 1

n− 1

)(
− λ

Tp

)mp

(7.10)

to obtain

λ

Ω
=

exp
(
− n

(∑
i αiλ

i + γλ
))
λsn−

∑
q nq

λn−1−
∑

p mp(eβλ + ρ)n

=
∑
m≥0

(−n)m

m!

λ

Ω
=

(∑
i αiλ

i + γλ
)m

λn−sn−1−
∑

p mp+
∑

q nq (eβλ + ρ)n

=
∑
m≥0

(−n)m

m!

m∑
ℓ=0

(
m

ℓ

)
γm−ℓ

λ

Ω
=

(∑
i αiλ

i
)ℓ

λm−ℓ

λn−sn−1−
∑

p mp+
∑

q nq (eβλ + ρ)n

=
∑
m≥0

(−n)m

m!

m∑
ℓ=0

∑
ℓ

(
m

ℓ

)(
ℓ

ℓ

)
αℓγm−ℓ

λ

Ω
=

1

λj((1 + ρ) + (eβλ − 1))n

=
∑
m≥0

(−n)m

m!

m∑
ℓ=0

∑
ℓ

(
m

ℓ

)(
ℓ

ℓ

)
αℓγm−ℓ

(1 + ρ)n

λ

Ω
=

1

λj(1 + (eβλ − 1)/(1 + ρ))n

=
∑
m≥0

(−n)m

m!

m∑
ℓ=0

∑
ℓ

(
m

ℓ

)(
ℓ

ℓ

)
αℓβjγm−ℓ

(1 + ρ)n

λ

Ω
=

1

(βλ)j(1 + (eβλ − 1)/(1 + ρ))n

using

1

(1 + (eβλ − 1)/(1 + ρ))n
=

∑
k≥0

(
k + n− 1

n− 1

)(
− eβλ − 1

1 + ρ

)k

(7.11)

and eβλ − 1 = O(λ) so that
λ

Ω
=

(eβλ − 1)k

λj
= 0

for k > j. Finally, the result follows observing that

λ

Ω
=

(eβλ − 1)k

(βλ)j
=

{ j

k

}k!

j!

so that the Omega variable λ is now eliminated. We additionally observe that the expansions used
in the proof in (7.10) and (7.11) must hold,∣∣ λ

Tp

∣∣, ∣∣eβλ − 1

1 + ρ

∣∣ < 1

and recalling that eα is entire. In the next section we will show that such conditions always hold
true by a suitable rescaling. □

7.2. Connection of Theorem 7.1 with previous work: Taylor series and integral rep-
resentations. We now show that several known results emerge as special cases of Theorem 7.1.
In the next proposition, we will need the Rodrigues formula for the Laguerre polynomials

L′(α) =
eα

n!
Dn

α

(αn

eα

)
. (7.12)



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 19

Proposition 7.2 ([56, Theorem 1]). The solution of

ζ = eξ
(ξ − τ)

(ξ − σ)

is given by the Taylor series

W0

( τ
σ
;0, 0, 1, ζ

)
= τ − S

∑
n≥1

(ζe−τ )n

n
L′(nS),

where S = τ − σ.

Proof. In this case α = 0, β = 0 = ρ, and γ, r, s = 1. Since αℓ = 0ℓ so that ℓ = 0 which implies
ℓ = 0 and βj = 0j so that j = 0. These observations imply

j = n−m+ ℓ−
∑
i

iℓi − sn− 1−
∑
p

mp +
∑
q

nq

=⇒ 0 = n−m+ 0− 0− n− 1− 0 + n1

=⇒ m = n1 − 1

and we obtain

W0

( τ
σ
;0, 0, 1, ζ

)
= τ +

∑
n≥1

(ζe−τ )n

n

n∑
n1=1

(
n

n1

)
(−n)n1−1

(n1 − 1)!
Sn1

= τ −
∑
n≥1

(ζe−τ )n

n2

n∑
n1=1

(
n

n1

)
(−n)n1

(n1 − 1)!
Sn1 .

In the expression above, we put m in place of n1. The result now follows by showing that

L′(nS) = −
n∑

k=1

(
n

k

)
(−nS)k−1

(k − 1)!
.

Indeed, using Rodrigues formula for the Laguerre polynomials in (7.12) and recalling the Omega
representation of the derivative in the proof of Lemma 2.2 we have

L′(α) =
eα

n!
Dn

α

(αn

eα

)
= eα

λ

Ω
=

(α+ λ)n

λneα+λ
=

λ

Ω
=

(α+ λ)n

λneλ

which implies

L′(α) = n
λ

Ω
=

(α+ λ)n−1

λneλ

=

n∑
m=0

n−1∑
ℓ=0

λ

Ω
=

(−λ)m

m!

n

λn

(
n− 1

ℓ

)
αn−1−ℓλℓ

=

n∑
m=0

n−1∑
ℓ=0

(−1)m

m!
n

(
n− 1

ℓ

)
αn−1−ℓ

λ

Ω
=
λℓ+m−n︸ ︷︷ ︸
=δℓ,n−m

=

n∑
m=1

(−1)m

m!
n

(
n− 1

n−m

)
αm−1

= −
n∑

m=1

(
n

m

)
(−α)m−1

(m− 1)!

by noting that when m = 0 we have δℓ,n−m = δℓ,n = 0 since ℓ ∈ [n− 1]0. □

If we compare the proof given above with the one given in [56, Theorem 1] we see that ours is
much more direct in the sense of avoiding the search for a pattern to be confirmed later by the
principle of induction. Indeed, see the proof of [56, Theorem 1].



20 A. F. NETO EJDE-2025/90

Proposition 7.3 ([56, Theorem 6]). The solution of ζ = ξeξ + ρξ is given by

Wρ

(−
−
;0, 1, 0, ζ

)
=

ζ

1 + ρ
+

∑
n≥2

M
(n)
n−1

( 1

1 + ρ

) ζn

n!(1 + ρ)n
,

where

M (m)
n (ζ) =

n∑
ℓ=1

mℓ
{n

ℓ

}
(−ζ)ℓ.

Proof. In this case α = 0, β = 1, and γ = 0 = τ1 (and hence δ = 0). Since αℓ = 0ℓ so that ℓ = 0
which implies ℓ = 0 and γm−ℓ = 0m−ℓ so that m = ℓ. These observations imply

j = n−m+ ℓ−
∑
i

iℓi − sn− 1−
∑
p

mp +
∑
q

nq (7.13)

=⇒ j = n−m+m− 0− 0n− 1− 0 + 0 (7.14)

=⇒ j = n− 1. (7.15)

Therefore, (
k + n− 1

n− 1

){ j

k

}k!

j!
=

(k + n− 1)!

k!(n− 1)!

{n− 1

k

} k!

(n− 1)!
=

nk

(n− 1)!

{n− 1

k

}
using (7.5), (7.6), and (7.13) so that the result follows by observing that

Wρ

(−
−
;0, 1, 0, ζ

)
=

∑
n≥1

n−1∑
k=0

ζn

n

(
k + n− 1

n− 1

)
(−1)k

{n− 1

k

} k!

(n− 1)!

1

(1 + ρ)k+n

=
ζ

1 + ρ
+

∑
n≥2

( n−1∑
k=0

nk

(n− 1)!

{n− 1

k

} (−1)k

(1 + ρ)k

) ζn

n!(1 + ρ)n

which implies the claim. □

Proposition 7.4 ([56, Theorem 9]). The solution of ζ = eξ(ξ− τ1)(ξ− τ2) is given by the Taylor
series

W0

(τ1 τ2
−

;0, 0, 1, ζ
)
= τ1 −

∑
n≥1

ζn

n!

(
− e−τ1

T

)n ∑
m≥0

nn−1(n+m− 1)!

m!(n−m− 1)!

( 1

nT

)m

,

where T = τ1 − τ2.

Proof. In this case α = 0, β, ρ, s = 0, γ = 1, and r = 2. Since αℓ = 0ℓ so that ℓ = 0 which
implies ℓ = 0 and βj = 0j so that j = 0. In this way we have

j = n−m+ ℓ−
∑
i

iℓi − sn− 1−
∑
p

mp +
∑
q

nq

=⇒ 0 = n−m+ 0− 0− 0n− 1−m2 + 0

=⇒ m = n− 1−m2

so that

W0

(τ1 τ2
−

;0, 0, 1, ζ
)
= τ1 +

∑
n≥1

n−1∑
m2=0

(−n)n−m2−1

(n−m2 − 1)!

(ζe−τ1)n

n

× (−1)m2

(
m2 + n− 1

n− 1

)
1

Tm2+n
2

and the result follows. (In the proposition statement we write m (T ) instead of m2 (T2).) □

Proposition 7.5 ([12, Theorem 1.1]). The solution of

ζ = ξeξ
∏
p

(ξ + Tp)



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 21

is given by the Taylor series

W0

( τ

−
;0, 0, 1, ζ

)
=

∑
n≥1

(−n)n−1

n!
T−n
2 . . . T−n

r

×
∑

m≥0r−1

(1− n)
∑

p mpnm2 . . . nmr

m2! . . .mr!(−nT2)m2 . . . (−nTr)mr
.

Proof. In this case α = 0, β, ρ, τ1 = 0, γ = 1, and s = 0. Since αℓ = 0ℓ so that ℓ = 0 which
implies ℓ = 0 and βj = 0j so that j = 0. In this way we have

j = n−m+ ℓ−
∑
i

iℓi − sn− 1−
∑
p

mp +
∑
q

nq

=⇒ 0 = n−m+ 0− 0− 0n− 1−
∑
p

mp + 0

=⇒ m = n− 1−
∑
p

mp

and the result follows from

W0

( τ

−
;0, 0, 1, ζ

)
=

∑
n≥1

∑
m≥0r−1

(−n)n−1−
∑

p mp(
n− 1−

∑
p mp

)
!

(ζe−τ )n

n

× (−1)
∑

p mp
∏
p

(
mp + n− 1

n− 1

)
1

T
mp+n
p

after observing that

(−1)
∑

p mp

n
(
n− 1−

∑
p mp

)
!
=

(n− 1)!(−1)
∑

p mp

n!
(
n− 1−

∑
p mp

)
!
=

(1− n)
∑

p mp

n!

and (
m+ n− 1

n− 1

)
=

nm

m!

using (7.5). □

The next proposition describes a generalized Lambert W function appearing in the context of
plane-symmetric Einstein-Maxwell fields.

Proposition 7.6 ([58, Theorem 3]). The solution of ζ = ξeαξ
2+ξ is given by the Taylor series

W0

(−
−
; (α,0d−2), 0, 1, ζ

)
=

1

α

∑
n≥1

(αζ)n

n

n−1∑
m=⌊(n−1)/2⌋

(
m

n−m− 1

)
(−n/α)m

m!
.

Proof. In this case β, ρ, τ1 = 0, γ = 1, and r = 0 = s. Since βj = 0j we have j = 0. It follows
that

j = n−m+ ℓ−
∑
i

iℓi − sn− 1−
∑
p

mp +
∑
q

nq

=⇒ 0 = n−m+ ℓ− 2ℓ2 − 0n− 1− 0 + 0

=⇒ 0 = n−m− ℓ− 0n− 1− 0 + 0

=⇒ ℓ = n−m− 1

since ℓ = ℓ2, because ℓi = 0, i ∈ [3, d], and the result follows by observing that

W0

(−
−
; (α,0d−2), 0, 1, ζ

)
=

∑
n≥1

∑
m≥0

m∑
ℓ=0

ζn

n

(−n)m

m!

(
m

ℓ

)
αℓ.

Finally, if ℓ = 0, then m = n− 1 and if ℓ = m, then m = n−m− 1 so that m = ⌊(n− 1)/2⌋. □



22 A. F. NETO EJDE-2025/90

Finally, we turn to obtaining previous integral representations. In contrast to Example 3.3,
there is a right-side enlargement in the interval of validity because of the integral representation
as discussed in [47].

Proposition 7.7 ([47, Equations (86) and (89)]). For ζ ∈ (−1/e, e) we have

W (ζ) = − 1

2π

∫ π

−π

eıθ ln
(
1− ζe−ıθ exp

(
− eıθ

))
dθ.

Proof. Using observation (2.2) in (7.7) we obtain

W (ζ) = −
λ

Ω
=
λ ln

(
1− ζλ−1e−λ

)
= − 1

2π

∫ π

−π

eıθ ln
(
1− ζe−ıθ exp

(
− eıθ

))
dθ.

□

Proposition 7.8 ([47, Equation (81)]). For ζ ∈ (−1/e, e) we have

W (ζ) =
1

2π

∫ π

−π

eıθ
(
eıθ + 1

)
1− ζe−ıθ exp

(
− eıθ

)dθ.
Proof. Using observation (2.2) in (7.8) we obtain

W (ζ) =
λ

Ω
=

λ(λ+ 1)

1− ζλ−1e−λ
=

1

2π

∫ π

−π

eıθ
(
eıθ + 1

)
1− ζe−ıθ exp

(
− eıθ

)dθ.
□

Proposition 7.9 ([47, Theorem 8.1]). If −1/e ≤ ζ ≤ e we have

W (ζ) =
1

π
ℜ
∫ π

0

ln
(exp (eıθ)− ζe−ıθ

exp
(
eıθ

)
− ζeıθ

)
dθ.

Proof. We first prove that

mn

n!
=

2

π
ℑ
∫ π

0

exp
(
meıθ

)
sin(nθ)dθ.

Note that

mn

n!
δϵ,ε =

λ

Ω
=

emλϵ

λεn
=⇒

λ

Ω
=

(
emλ − em/λ

)λn − λ−n

2
= −mn

n!

=⇒ 1

ı

λ

Ω
=

(
emλ − em/λ

)λn − λ−n

2ı
=

mn

n!

=⇒ 1

2πı

∫ π

−π

(
exp

(
meıθ

)
− exp

(
me−ıθ

))
sin(nθ)dθ =

mn

n!

which is equivalent to the desired result. The result now follows using (2.2) in (7.7) and using the
argument in [57, Section 1.7.3] showing that the representation is not valid when ζ < −1/e and
ζ > e. □

Remark 7.10. There are other cases that fit the general representation given in Theorem 3.7
although this may not appear to be the case at first sight. The reason is because under simple
transformations they are related to W (ζ). Indeed, see [38, 39] and also the application discussed
in Section 6, more precisely, Proposition 9.2.

8. Radius of convergence for the Taylor series of the generalized Lambert W
function

We show that we can improve upon certain aspects the radius of convergence mentioned in [56,
Theorems 2 and 9]. In particular, we aim to address some of the open questions left open in [56].
For the convenience of the reader the open questions are quoted verbatim below:

Q2: Theorems 1 and 9 contain Taylor series including derivatives of Laguerre polynomials, and
the Bessel polynomials. Could we say more on these Taylor series, especially how to find
the radius of convergence in general apart from Theorem 2?



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 23

Q3: We could say nothing about the radius of convergence of (14) for the r-Lambert function
except the only case r = −2. What can we say about the convergence radius ρr in general?

Our strategy consists in using as a starting point Theorem 7.1.

8.1. Main theorem. We can take |β| << 1 and |τp| >> 1 so that |Tp| >> 1 by using a suitable
rescaling ξ → ξ/α with α sufficiently large so that

exp
(∑

i

αiξ
i/αi

)
(exp(βξ/α) + ρ) exp(γξ/α+ δ)

ξ
∏

p(ξ + αTp)∏
q(ξ + αSq)

= αr−sζ

and in all the expressions that follow Definition 2.1 applies. We are now ready to state the main
result of this section.

Theorem 8.1. The radius of convergence of the power series representation for

Wρ

( τ
σ
;α, β, γ, ζ

)
is

R = eδ−1|1 + ρ|
∏

p |Tp|∏
q |Sq|

∣∣∣ β

1 + ρ
+ γ −

∑
q

1

Sq
+

∑
p

1

Tp

∣∣∣−1

.

Proof. We write

⟨ζn⟩Wρ

( τ
σ
;α, β, γ, ζ

)
= γn.

Next, we define

A(λ) :=
exp

(
− n

∑
i αiλ

i − nγλ
)∏

q S
n
q (1 + λ/Sq)

n∏
p T

n
p (1 + λ/Tp)n

so that

γn =
e−nδ

n

λ

Ω
=

A(λ)

λn−1(1 + ρ+ exp(βλ)− 1)n

=
e−nδ

n

λ

Ω
=

A(λ)

λn−1(1 + ρ)n(1 + (exp(βλ)− 1)/(1 + ρ))n

=
e−nδ

n

λ

Ω
=

nn−1A(λ/n)

λn−1(1 + ρ)n(1 + (exp(βλ/n)− 1)/(1 + ρ))n

≈ e−nδ

n

λ

Ω
=

nn−1A(λ/n)

λn−1(1 + ρ)n(1 + βλ/(n(1 + ρ)))n

≈ e−nδ

n

λ

Ω
=

nn−1 exp(−βλ/(1 + ρ))A(λ/n)

λn−1(1 + ρ)n
,

where the third equality follows from the invariance under rescaling λ → λ/n and the last one
follows using (3.5). Next, we observe that

A(λ/n) =
exp

(
− n

∑
i αiλ

i/ni − γλ
)∏

q S
n
q

(
1 + λ/(nSq)

)n∏
p T

n
p (1 + λ/(nTp))n

≈
exp(−γλ)

∏
q S

n
q

(
1 + λ/(nSq)

)n∏
p T

n
p

(
1 + λ/(nTp)

)n
≈

∏
q S

n
q∏

p T
n
p

exp
(
−
(
γ −

∑
q

1

Sq
+

∑
p

1

Tp

)
λ
)

using again invariance under rescaling λ → λ/n and observing that

exp
(
− n

∑
i

αiλ
i/ni

)
≈ 1



24 A. F. NETO EJDE-2025/90

along with (3.5). Finally, we have

γn ≈
e−nδnn−1

∏
q S

n
q

n
∏

p T
n
p (1 + ρ)n

λ

Ω
=

exp
(
− (β/(1 + ρ) + γ −

∑
q 1/Sq +

∑
p 1/Tp)λ

)
λn−1

≈
e−nδnn−1

∏
q S

n
q

n
∏

p T
n
p (1 + ρ)n

(−1)n−1

(n− 1)!

( β

1 + ρ
+ γ −

∑
q

1

Sq
+

∑
p

1

Tp

)n−1

≈
e−nδnn−1

∏
q S

n
q∏

p T
n
p (1 + ρ)n

(−1)n−1

n!

( β

1 + ρ
+ γ −

∑
q

1

Sq
+

∑
p

1

Tp

)n−1

to obtain

R = lim
n→∞

∣∣ γn
γn+1

∣∣ = eδ−1|1 + ρ|
∏

p |Tp|∏
q |Sq|

∣∣∣ β

1 + ρ
+ γ −

∑
q

1

Sq
+

∑
p

1

Tp

∣∣∣−1

using Example 3.3. □

8.2. Connection of Theorem 8.1 with previous work: convergence radius of certain
Taylor series. We first observe that if we take τ1 = 0 = ρ, γ = 1 = s, r = 2, and S1 = T2 in
Theorem 8.1 we obtain the well-known radius of convergence of the usual W function as discussed
in Example 3.3. Our answer to the query Q2 is the content of the next corollary.

Corollary 8.2. The radius of convergence of the power series for W0

(
τ
σ ;0, 0, γ, ζ

)
is

R = eδ−1

∏
p |Tp|∏
q |Sq|

∣∣γ −
∑
q

1

Sq
+

∑
p

1

Tp

∣∣−1
.

Remark 8.3. The radius of convergence R stated in Theorem 8.1 improves the previous result
[56, Theorem 2] in the sense that it is valid for all S = τ − σ ̸= 0 and not only if S < 0, but
the radius of convergence in [56, Theorem 2] improves ours if S < 0. As a way of comparison, by
adapting to our notation, [56, Theorem 2] reads

R(MB) = exp
(τ + σ

2
− 2

√
−S

)
,

where the superscript (MB) stands for Mező-Baricz and note that −S > 0 while our result reads

R(N) =
eτ−1

|S|
∣∣1− 1

S

∣∣−1
=

eτ−1

|1− S|
,

where the superscript (N) stands for Neto. Furthermore, note also that if T = −S > 0 we have

R(MB) = exp
(
− T

2
− 2

√
T + σ

)
and

R(N) =
e−T−1+σ

|1 + T |
such that R(N) < R(MB) if T > 0.

Remark 8.4. The radius of convergence R stated in Theorem 8.1 improves the previous result
[56, Theorem 9] in the sense that no radius of convergence was obtained in [56, Theorem 9].
Indeed, we recall the verbatim commentary taken from [56]: “Neither Mugnaini nor the authors
could calculate the radius of convergence of this series.”

We finally address the open problem quoted verbatim above in Q3. For easy of reference we
have the correspondence between notations

r(MB) = ρ(N) and ρ(MB)
r = R(N).

The ρ-Lambert function is the solution ξ = Wρ(ζ) of the equation

ξeξ + ρξ = ζ.

Our answer is the content of the next corollary.



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 25

Corollary 8.5. The radius of convergence of the Taylor series of Wρ(ζ) around 0 is

R =
∣∣ (1 + ρ)2

e

∣∣.
9. A sequence arising in algebraic geometry

The problem comprises when solving the functional equation

a(1 + χa) ln(1 + χa) = (a+ 1)χa − ζ (9.1)

which arises in the study of configuration spaces [52, Chapter 4, Equation (4.25)] and the exact
determination of χa is related to WDVV equations in physics [62]. We write

χa(ζ) = ζ +
∑
n≥2

mn(a)
ζn

n!
.

We consider first the simpler problem of solving

(1 + χ−) ln(1 + χ−) = ζ. (9.2)

It is easy to show that (9.2) is equivalent to

W (ζ)eW (ζ) = ζ.

under the observation χ−(ζ) = eW (ζ) − 1. In words, the solution of our original problem; that is,
determine the solution of (9.2) is now connected to the Lambert W function. Therefore, we can
write

χ−(ζ) = eW (ζ) − 1 = (F ◦W )(ζ)

with F (ζ) = eζ − 1 so that Theorem 3.1 can be applied as the next proposition shows.

Proposition 9.1 ([32, Theorem 2.4]). We have

mn(−1) = (1− n)n−1.

Proof. Note that (
F ◦G⟨−1⟩)(ζ) = α0 −

λ

Ω
=
λF ′(λ) ln

(
1− ζ

G(λ)

)
= −

λ

Ω
=
λeλ ln

(
1− ζ

λeλ

)
=

∑
n≥1

ζn

n

λ

Ω
=

e(1−n)λ

λn−1

=
∑
n≥1

ζn
(1− n)n−1

n!

using Theorem 3.1 with F (ζ) = eζ − 1 so that α0 = 0 and G⟨−1⟩ = W so that G(ζ) = ζeζ . □

More generally, we have the next result.

Proposition 9.2. We have

χa(ζ) = exp
(
α+W (−βζ − γ)

)
− 1,

where

α :=
a+ 1

a
, β :=

1

aeα
, and γ :=

α

eα
.

Proof. We introduce the change of variables

χa = eα+ξ − 1

to obtain
−(a+ 1)− aξeα+ξ = ζ =⇒ ξeξ = −βζ − γ

using (9.1) and the result follows by recalling the definition of the Lambert W function. □



26 A. F. NETO EJDE-2025/90

We next recall the definition of the generalized Stirling numbers of the second kind (see, e.g.,
[13, Chapter 5, Page 57]) ∑

m,n≥0

Sr(m,n)
ξnτm

m!
= exp

(
ξ
(
eτ −

r−1∑
k=0

τk

k!

))
. (9.3)

In combinatorial terms Sr(m,n) counts the number of partitions of [m] in n blocks with at least
r elements. Note that

S1(m,n) =
{m

n

}
in (7.4). For later use we also recall the fundamental triangular recurrence relation (see, e.g., [13,
Chapter 5, Page 57])

Sr(m+ 1, n) = nSr(m,n) +

(
m

r − 1

)
Sr(m− r + 1, n− 1). (9.4)

We can now state a generalization of a result of Koganov as stated in [32].

Theorem 9.3. We have

mn(a) = 1 + (n− 1)!

na∑
j=1

n(n+ 1) . . . (n+ j − 1)

n−1∑
ℓ=0

min(j,ℓ)∑
k=0

S1(ℓ+ 1, k + 1)

ℓ!

× ak(−1)j−k(a+ 1)j−k S2(n− ℓ− 1 + j − k, j − k)

(n− ℓ− 1 + j − k)!
,

where

na :=
⌊n− 1

2

⌋
δa,1 + (n− 1)(1− δa,1).

Proof. Note that (
F ◦G⟨−1⟩)(ζ) = α0 −

λ

Ω
=
λF ′(λ) ln

(
1− ζ

G(λ)

)
= −

λ

Ω
=
λeλ ln

(
1− ζ

(a+ 1)(eλ − 1) + aλeλ

)
= −

λ

Ω
=
λeλ ln

(
1− ζ/λ

1−Da(λ)

)
,

where

Da(λ) := a(eλ − 1)− (a+ 1)
eλ − 1− λ

λ

and using Theorem 3.1 with F (ζ) = eζ − 1 as in Proposition 9.2 so that we have again α0 = 0,
but now we take G(ζ) = (a+ 1)(eζ − 1)− aζeζ . In this way we obtain(

F ◦G⟨−1⟩)(ζ) = ∑
n≥1

ζn

n

λ

Ω
=

λeλ

λn(1−Da(λ))n
.

Next, we use
1

(1−Da(λ))n
=

∑
j≥0

n(n+ 1) . . . (n+ j − 1)

j!
Dj

a(λ). (9.5)

We next observe that Da(λ) = O
(
λ1+δa,1

)
so that we have j ≤ na, because

j(1 + δa,1) ≤ n− 1 =⇒ j ≤ n− 1

1 + δa,1

and
1

1 + δa,1
=

δa,1
2

+ (1− δa,1).



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 27

We next note that

Dj
a(λ) =

j∑
k=0

(
j

k

)
ak(eλ − 1)k(−1)j−k(a+ 1)j−k (e

λ − 1− λ)j−k

λj−k

to obtain

λ

Ω
=

eλ

λn−1

Dj
a(λ)

j!
=

j∑
k=0

1

j!

(
j

k

)
ak(−1)j−k(a+ 1)j−k

λ

Ω
=

eλ

λn−1+j−k

(eλ − 1)k

k!

(eλ − 1− λ)j−k

(j − k)!
.

Finally, we have

eλ(eλ − 1)k

k!
= (k + 1)

(eλ − 1)k+1

(k + 1)!
+

(eλ − 1)k

k!

to obtain

λ

Ω
=

eλ

λn−1+j−k

(eλ − 1)k

k!

(eλ − 1− λ)j−k

(j − k)!

=
λ

Ω
=

1

λn−1+j−k

(
(k + 1)

(eλ − 1)k+1

(k + 1)!
+

(eλ − 1)k

k!

) (eλ − 1− λ)j−k

(j − k)!

=
∑

ℓ+m=n−1+j−k

(k + 1)S1(ℓ, k + 1) + S1(ℓ, k)

ℓ!

S2(m, j − k)

m!

=
∑

ℓ+m=n−1+j−k

S1(ℓ+ 1, k + 1)

ℓ!

S2(m, j − k)

m!

using Lemma 2.5 with k → n− 1 + j − k,

F (λ) =
(eλ − 1)ℓ

ℓ!

such that ℓ = k, k + 1, and

G(λ) =
(eλ − 1− λ)j−k

(j − k)!

along with

Sr(m,n)

m!
=

1

n!

λ

Ω
=

1

λm

(
eλ −

r−1∑
k=1

λk

k!

)
which follows from (9.3) to obtain the second equality and (9.4) to obtain the last one. Collecting
the results above and observing that ℓ ≥ k and m ≥ j − k so that 0 ≤ ℓ ≤ n − 1 and 0 ≤ k ≤ p
with p = j, k so that 0 ≤ k ≤ min(j, ℓ) we arrive at the desired result by going back to (9.5). □

Corollary 9.4. We have

mn(1) = 1 + (n− 1)!

⌊n−1
2 ⌋∑

j=1

n(n+ 1) . . . (n+ j − 1)

n−1∑
ℓ=0

min(j,ℓ)∑
k=0

S1(ℓ+ 1, k + 1)

ℓ!

× (−2)j−k S2(n− ℓ− 1 + j − k, j − k)

(n− ℓ− 1 + j − k)!
.

This is nothing more than Koganov’s aforementioned expression apart from a minor sign cor-
rection: j − k should replace k − j in

S2(n− ℓ− 1 + k − j, j − k)

(n− ℓ− 1 + k − j)!

as stated in [32, Page 3].



28 A. F. NETO EJDE-2025/90

10. Higher order derivatives of F ◦G⟨−1⟩

In this section we generalize the main result in [42]. The exact determination of higher order
derivatives has some interesting consequences for the Langevin function [41]. In all that follows
we let Fk(ζ) = ζk.

Theorem 10.1. We have

Dm
ζ

(
Fk ◦G⟨−1⟩)(ζ) = k

m−k∑
ℓ=0

(β1)
−m−ℓ(−1)ℓRm−k,ℓ

(
β2, . . . , βm−k−ℓ+2

)
,

where

Rm−k,ℓ

(
β2, . . . , βm−k−ℓ+2

)
=

1

ℓ!

∑
k1+···+kℓ=m−k+ℓ

(m+ ℓ− 1)!βk1
. . . βkℓ

with ki ∈ [2,m− k − ℓ+ 2].

Proof. Using Theorem 3.1 we obtain

⟨ζm⟩
(
Fk ◦G⟨−1⟩)(ζ) = k

m

λ

Ω
=

λk

Gm(λ)

=
k

mβm
1

λ

Ω
=

1

λm−k

1

(1 + (G(λ)/(β1λ)− 1))m

=
k

mβm
1

λ

Ω
=

1

λm−k

m−k∑
ℓ=0

(−1)ℓ
(
ℓ+m− 1

m− 1

)(β2λ

β1
+ · · ·+ βm−k+1λ

m−k

β1

)ℓ

.

The result follows by using

λ

Ω
=

1

λm−k

(
β2λ+ . . . βm−k+1λ

m−k
)ℓ

=
λ

Ω
=

1

λm−k

(
β2λ+ · · ·+ βm−k−ℓ+2λ

m−k−ℓ+1
)ℓ

=
∑

(k1−1)+···+(kℓ−1)=m−k

βk1 . . . βkℓ

=
∑

k1+···+kℓ=m−k+ℓ

βk1 . . . βkℓ

with ki ∈ [2,m− k − ℓ+ 2]. By considering(
β2λ+ . . . βm−k+1λ

m−k
)ℓ ≡ (. . . )ℓ

we have λa with a ≤ m− k if ℓ = 1, a ≤ m− k − 1 if ℓ = 2 and so on to arrive at (. . . )ℓ so that
a ≤ m− k − (ℓ− 1) to obtain the first equality. The general term in(

β2λ+ · · ·+ βm−k−ℓ+2λ
m−k−ℓ+1

)ℓ
is of the form

∏ℓ
i=1 βki

λki−1 such that the Omega operator selects only terms satisfying

λ

Ω
=

∏ℓ
i=1 λ

ki−1

λm−k
= δ∑ℓ

i=1(ki−1),m−k

and the second equality follows. □

By setting k = 1 we obtain the corollary that follows. To connect with [42, Theorem 2] observe
that βk = G(k)(0)/k!.

Corollary 10.2 ([42, Theorem 2]). We have

Dm
ζ G⟨−1⟩(ζ) =

m−1∑
ℓ=0

(β1)
−m−ℓ(−1)ℓRm−1,ℓ

(
β2, . . . , βm−ℓ+1

)
,

where

Rm−1,ℓ

(
β2, . . . , βm−ℓ+1

)
=

1

ℓ!

∑
k1+···+kℓ=m−1+ℓ

(m+ ℓ− 1)!βk1
. . . βkℓ

with ki ≥ 2.



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 29

We can also compute higher order derivatives using Theorem 3.7. Indeed, we first observe that
the Omega representation of F ◦G⟨−1⟩ carries the term

λeF (λ) det
(
DiGj(λ)

)
∈ C[[λ]].

(See the right-hand side of (3.7).) Therefore, if we consider λk with k = (k1, . . . , kn) and use the

notation Fk(ζ) = ζk, then it is possible to determine Dm
ζ

(
F ◦G⟨−1⟩)(ζ) by computing the simpler

expression Dm
ζ

(
Fk ◦G⟨−1⟩)(ζ).

Theorem 10.3. We have

Dm
ζ

(
Fk ◦G⟨−1⟩)(ζ) = 1

βm+e

n∏
k=1

∑
ℓk

(−1)ℓk
(
ℓk +mk

mk

)∑
β1k⃗1

. . . βnk⃗n
,

where k⃗j = (kj1, . . . ,kjℓj ) and the sum is taken subject to

ℓ1∑
i1=1

k1i1 + · · ·+
ℓn∑

in=1

knin = m− k+ e.

Proof. Using Theorem 3.7 we have

⟨ζm⟩
(
Fk ◦G⟨−1⟩)(ζ)

=
λ

Ω
=

λk∏n
k=1(Gk(λ))mk+1

=
1

βm+e

λ

Ω
=

1

λm−k+e ∏n
k=1(1 +Hk(λ))mk+1

=
1

βm+e

n∏
k=1

∑
ℓk

(−1)ℓk
(
ℓk +mk

mk

)
λ

Ω
=

(Hk(λ))
ℓk

λm−k+e

=
1

βm+e

n∏
k=1

∑
ℓk

(−1)ℓk
(
ℓk +mk

mk

)
λ

Ω
=

(∑
k1

β1k1λ
k1
)ℓ1

. . .
(∑

kn
βnknλ

kn
)ℓn

λm−k+e

=
1

βm+e

n∏
k=1

∑
ℓk

(−1)ℓk
(
ℓk +mk

mk

)∑
β1k⃗1

. . . βnk⃗n
,

where k⃗j = (kj1, . . . ,kjℓj ) such that βjk⃗j
= βjkj1

. . . βjkjℓj
with j ∈ [n] and we write(∑

kj

βjkjλ
kj

)ℓj
=

(∑
kj1

βjkj1λ
kj1

)
. . .

(∑
kjℓj

βjkjℓj
λkjℓj

)
for each j. The action of the Omega operator results in the constraints: the sum

∑
ℓk

is limited

to maxj∈[n](mj − kj + 1) and the
∑

is taken subject to

ℓ1∑
i1=1

k1i1 + · · ·+
ℓn∑

in=1

knin = m− k+ e.

□

11. Concluding Remarks

Up to now, there are several known proofs of the L-B inversion formula with several different
flavors such as combinatorial, algebraic, analytic, and so on, and this work adds another perspective
on this topic. We showed that an OC based proof of the L-B inversion formula is available in
Theorems 3.1, 3.5, and 3.7 stated in Section 3. In a certain sense, this work comprises another
instance of computing the continuous discretely [5]. By this we mean recasting the L-B theorem to
compute inverse functions (hence a continuous tool) via combinatorial analysis using the OC which,
as already mentioned, was originally used to study diophantine equations (hence a discrete tool).



30 A. F. NETO EJDE-2025/90

This line of approach already made its appearance before and this work provides another instance
where tools developed to address discrete problems are also useful in the continuous scenario. One
may therefore ask what is the point of our OC based representation of the L-B formula? We
have several answers to the query. First, our proofs are simple and our representation has the
nice feature of requiring simple elimination rules in the context of the OC to obtain free Omega
generating functions because all the expressions involved contain factored polynomials. Second, as
one may see by carefully reading Sections 4, 5, and 6 we have at least two available proofs of most
results regarding the aforementioned versions of the L-B inversion formula showing the versatility
of the method in a way that the Omega elimination rules play and essential role. Third, some
applications are advanced to show the versatility and usefulness of the aforementioned approach.
In particular, we introduced a new generalized Lambert W function in Theorem 7.1 which implies
several previous known theorems scattered in the literature and stated as propositions in Section 7.
Furthermore, it provides insight into some of the open problems stated in [56] which are answered
using our approach in Sections 7 and 8 dealing with the Omega free expansion of the generalized
W function and the radius of convergence of some subclasses of the aforementioned generalized
W function, respectively. We highlight also that the OC representation was crucial in obtaining
by simple means the convergence radius of some generalized Lambert W functions in Section 8
pointing out another instance outside the original scope where OC can be useful. In Section 9
a generalization of a representation of a sequence arising in algebraic geometry was determined
which implies Koganov’s representation as a special case [32]. Finally, in Section 10 we showed that
higher order derivatives of F ◦G⟨−1⟩ can be easily handled using OC based methods generalizing
some previous results in the literature [42] corresponding to the case F ≡ I and G ≡ G.

Recently OC has shown to be extremely useful not only to describe the partition of natu-
ral numbers, but in a number of other problems including inverse problems such as the inverse
problem for non-autonomous dynamical systems (answering a question left open in [4]) and the
L-B inversion formula along with the generalized Lambert W function with this work (answering
some of the questions left open in [56]). Therefore, this work reinforces the ubiquitous nature
and usefulness character of OC at the interplay of the discrete-continuous settings by exploring
the computation of inverse functions and associated consequences, most notably the case of the
generalized Lambert W function.

References

[1] A. Abdesselam; A physicist’s proof of the Lagrange-Good multivariable inversion formula. J. Phys. A, 36
(2003), 9471–9478.

[2] G. E. Andrews, P. Paule, A. Riese; MacMahon’s partition analysis: the omega package. European J. Combin.,
22 (2001), no. 7, 887–904

[3] G. E. Andrews, P. Paule, A. Riese; MacMahon’s partition analysis VI: a new reduction algorithm. Ann. Comb.,

5 (2001), no. 3-4, 251–270.
[4] A. Bassom, P. Howlett, P. Taylor; An explicit Maclaurin series solution to a classic non-autonomous abstract

evolution equation. Appl. Math. Lett., 139 (2023), 108537.

[5] M. Beck, S. Robins; Computing the continuous discretely: integer-point enumeration in polyhedra. Vol. 2,
Springer, New York, 2007.

[6] J. A. Bondy, U. S. R. Murty; Graph theory. Springer, 2008.

[7] M. Borinsky; Graphs in perturbation theory: algebraic structure and asymptotics. Springer, Switzerland, 2018.
[8] C. Brouder, G. H. E. Duchamp, F. Patras, G. Z. Toth; The Rayleigh-Schrödinger perturbation series of quasi-

degenerate systems. Int. J. Quantum Chem., 112 (2012), no. 10, 2256–2266.

[9] C. Brouder, A. Mestre, F. Patras; Tree expansion in time-dependent perturbation theory. J. Math. Phys., 51
(2010), 072104.

[10] F. Carbonell, J. Jimenez, L. Pedroso; Computing multiple integrals involving matrix exponentials. J. Comput.
Appl. Math., 213 (2008), 300–305.

[11] T. Carletti; The Lagrange inversion formula on non-Archimedean fields, non-analytical form of differential and

finite difference equations. Discrete Contin. Dyn. Syst., 9 (2003), no. 4, 835–858.
[12] P. Castle; Taylor series for generalized Lambert W functions. arXiv:1801.09904 (2018).

[13] L. Comtet; Analyse Combinatorie, Vol. II, Presses Universitaires de France, Paris, 1970.

[14] F. Constantinescu; Perturbation expansion for distributions on surfaces, Lagrange-Bürmann theorem and
applications to mechanics. Generalized Functions and Their Applications, (1993), 51–52.

[15] H. D. Dahmen, W. Krzyzanowski, M. L. Larsen; Graph rules for functions of the time-evolution operator.

Phys. Rev. D, 33 (1986), 1726–1744.



EJDE-2025/90 LAGRANGE-BÜRMANN THEOREM VIA OMEGA CALCULUS 31

[16] R. M. Digilov; Gravity discharge vessel revisited: an explicit Lambert W function solution. American Journal

of Physics, 85 (2017), no. 7, 510–514

[17] F. J. Dyson; The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev., 75 (1949), no. 3,
486–502.

[18] A. Francisco Neto; Matrix analysis and omega calculus. SIAM Rev., 62 (2020), no. 1, 264–280.

[19] A. Francisco Neto; Matrix computations with the omega calculus. Linear Multilinear Algebra, 70 (2022), no.
20, 5075–5106.

[20] A. Francisco Neto; A basis- and integral-free representation of time-dependent perturbation theory via the

Omega matrix calculus. Ann. Inst. Henri Poincaré Comb. Phys. Interact., 11 (2024), 383–407.
[21] A. Francisco Neto; A new approach to multi-delay matrix valued fractional linear differential equations with

constant coefficients. Fract. Calc. Appl. Anal., 26 (2023), no. 5, 2202–2236.

[22] A. Francisco Neto; An explicit Maclaurin series solution to non-autonomous and non-homogeneous evolution
equation, Omega Calculus, and associated applications, IMA J. Appl. Math., 89 (2024), no. 3, 533–573.

[23] A. Francisco Neto; A new approach to MacMahon’s partition analysis and associated applications. Linear
Multilinear Algebra, 73 (2025), no. 6, 1216–1256.

[24] A. Francisco Neto; A new basis- and integral-free approach to perturbation theory: the Schrodinger dynamics

of N trapped ions in the high-intensity regime. Ann. Physics, 474 (2025), 16997.
[25] H. M. Fried; Green’s functions and ordered exponentials. Cambridge University Press, Cambridge, 2002.

[26] A. M. Garsia; A q-analogue of the Lagrange inversion formula. Houston J. Math, 7 (1981), no. 2, 205–237.

[27] I. M. Gessel; A combinatorial proof of the multivariable Lagrange inversion formula. J. Combin. Theory Ser.
A, 45 (1987), no. 2, 178–195.

[28] I. M. Gessel; A noncommutative generalization and q-analog of the Lagrange inversion formula. Trans. Amer.

Math. Soc., 257 (1980), no. 2, 455–482
[29] I. M. Gessel; Lagrange inversion. J. Combin. Theory Ser. A, 144 (2016), 212–249.

[30] P.-L. Giscard, K. Lui, S. J. Thwaite, D. Jaksch; An exact formulation of the time-ordered exponential using

path-sums. J. Math. Phys. 56 (2015), no. 5, 053503.
[31] I. J. Good; Generalizations to several variables of Lagrange’s expansion, with applications to stochastic pro-

cesses. Math. Proc. Cambridge Philos. Soc., 56 (1960), no. 4, 367–380.
[32] I. P. Goulden, S. Litsyn, V. Shevelev; On a sequence arising in algebraic geometry. J. Integer Seq., 8.05.4

(2005), 7.

[33] G.-H. Han; A general algorithm for the MacMahon omega operator. Ann. Comb. 7 (2003), 467–480.
[34] P. Henrici; An algebraic proof of the Lagrange-Bürmann formula. J. Math. Anal. Appl., 8 (1964), no. 2,

218–224.

[35] P. Henrici; Topics in Computational Complex Analysis: IV. The Lagrange-Bürmann Formula for Systems of
Formal Power Series. Computational Aspects of Complex Analysis: Proceedings of the NATO Advanced Study

Institute held at Braunlage, Harz, Germany, July 26–August 6, 1982. Dordrecht: Springer Netherlands, 1983.

[36] P. Henrici; Applied and computational complex analysis, Vol. 1. Wiley, New York, 1974.
[37] J. Hofbauer; A short proof of the Lagrange-Good formula. Discrete Math. 25 (1979), no. 2, 135–139.

[38] A. Houari; New analytical results in solid state physics using the Lambert W function. European Journal of

Physics 44 (2023), no. 6, 065502.
[39] A. Houari; Additional applications of the Lambert W function to solid state physics. European Journal of

Physics, 45 (2024), no. 5, 055501.
[40] I-C. Huang; Residue methods in combinatorial analysis. In: Local cohomology and its applications, edited by

G. Lyubeznik, pp. 255–352, Marcel Dekker Inc., 2002.

[41] M. Itskov, R. Dargazany, K. Hörnes; Taylor expansion of the inverse function with application to the Langevin
function. Math. Mec. Solids 17 (2012), no. 7, 693–701.

[42] W. P. Johnson; Combinatorics of higher derivatives of inverses. Amer. Math. Monthly 109 (2002), no. 3,
273–277.

[43] G. W. Johnson, M. L. Lapidus; The Feynman integral and Feynman’s operational calculus. Oxford University
Press, Oxford, 2002.

[44] S. A. Joni; Lagrange inversion in higher dimensions and umbral operators. Linear Multilinear Algebra 6 (1978),
no. 2, 111–121.

[45] A. Kalev, I. Hen; An integral-free representation of the Dyson series using divided differences. New J. Phys.,
23 (2021), 103035.

[46] A. Kalev, I. Hen; Feynman path integrals for discrete-variable systems: walks on Hamiltonian graphs. Phys.
Rev. Res. 7 (2025), 013220.

[47] G. A. Kalugin, D. J. Jeffrey, R. M. Corless; Stieltjes, Poisson and other integral representations for functions
of Lambert W . arXiv preprint arXiv:1103.5640 (2011).

[48] Ch. Krattenthaler; Operator methods and Lagrange inversion: a unified approach to Lagrange formulas. Trans.
Amer. Math. Soc., 305 (1988), no. 2, 431–465.

[49] I. Lindgren, S. Salomonson, B. Åsén; The covariant-evolution-operator method in bound-state QED. Phys.

Rep., 389 (2004), no. 4, 161–261.
[50] H. Maassen, D. Botvich; A Galton–Watson estimate for Dyson series. Ann. Henri Poincaré, 10 (2009), 1141–

1158



32 A. F. NETO EJDE-2025/90

[51] P. A. MacMahon; Combinatory Analysis. Volumes I and II, 137, AMS Chelsea Publishing, Providence, 2001

[52] Y. I. Manin; Frobenius manifolds, quantum cohomology, and moduli spaces. Vol. 47. American Mathematical

Soc., 1999.
[53] J. Mart́ınez-Carranza, F. Soto-Eguibar, H. Moya-Cessa; Alternative analysis to perturbation theory in quantum

mechanics: Dyson series in matrix form. The European Physical Journal D, 66 (2012), 1–6.

[54] V. P. Maslov et al.; Operational methods. Mir Publishers, Moscou, 1976.
[55] D. Merlini, R. Sprugnoli, M. Cecilia Verri; Lagrange inversion: when and how. Acta Appl. Math., 94 (2006),

no. 3, 233–249.

[56] I. Mező, A. Baricz; On the generalization of the Lambert W function. Trans. Amer. Math. Soc. 369 (2017),
no. 11, 7917–7934.

[57] I. Mező; The Lambert W function: its generalizations and applications. Chapman and Hall/CRC, 2022.

[58] I. Mező, C. B. Corcino, R. B. Corcino; Resolution of the plane-symmetric Einstein-Maxwell fields with a
generalization of the Lambert W function. Journal of Physics Communications, 4 (2020), no. 8, 085008.

[59] I. Mező, G. Keady; Some physical applications of generalized Lambert functions. European Journal of Physics
37 (2016), no. 6, 065802.

[60] J.-C. Novelli, J.-Y. Thibon; Noncommutative symmetric functions and Lagrange inversion. Adv. Appl. Math.

40 (2008), no. 1, 8–35.
[61] E. Panzer, R. Wulkenhaar; Lambert-W solves the noncommutative Φ4-model. Comm. Math. Phys. 374 (2020),

no. 3, 1935–1961.

[62] M. A. Readdy; The pre-WDVV ring of physics and its topology. Ramanujan J. 10 (2005), 269–281.
[63] M. Reed, B. Simon; Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic

Press, San Diego, 1975.

[64] J. Santmayer; Problems and solutions, Math. Mag., 97, no. 4, 433–443.
[65] A. Schulze-Halberg, A. M. Ishkhanyan; Exactly-solvable quantum systems in terms of Lambert-W functions.

Few-Body Systems 61 (2020), 1–15.

[66] T. C. Scott, R. Mann, R. E. Martinez II; General relativity and quantum mechanics: towards a generalization
of the Lambert W function. Appl. Algebra Engrg. Comm. Comput., 17 (2006), 41–47.

[67] A. Sokal; A ridiculously simple and explicit implicit function theorem. Sém. Lothar. Combin., 61A
(2009/2011), Article B61Ad

[68] A. Tavlayan, B. Tekin; Exact formulas for spherical photon orbits around Kerr black holes. Phys. Rev. D, 102

(2020), no. 10, 104036.
[69] L. Verde-Star; Dual operators and Lagrange inversion in several variables. Adv. Math. 58 (1985), no. 1, 89–108.

[70] M. Vittot; Perturbation theory and control in classical or quantum mechanics by an inversion formula. J. Phys.

A, 37 (2004), no. 24, 6337–6358.
[71] G. Xin; A Euclid style algorithm for MacMahon’s partition analysis. J. Combin. Theory Ser. A, 131 (2015),

32–60.

[72] S. Yi, P. W. Nelson, A. G. Ulsoy; Time-delay systems: analysis and control using the Lambert W function.
World Scientific, Singapore, 2010.

Antônio Francisco Neto

UFOP, DEPRO-Escola de Minas, Campus Morro do Cruzeiro, 35.400-000, Brazil
Email address: antonio.neto@ufop.edu.br