In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. These tools are subsequently applied to derive counterparts for the exponential identity $e^{A+B} = e^A e^B$ and the binomial theorem $(A+B)^n = \sum \binom{n}{k} A^k B^{n-k}$ when the commutator $[B, A]$ is either an arbitrary quadratic polynomial or a monomial in $A$ or $B$. Analogous formulas are found when the commutator is bivariate. Furthermore, we introduce a novel operator bridging between the normal and antinormal forms.

§ INTRODUCTION

The process of normal ordering is known as a challenging subject, as the discoveries are often only partial results. Numerous contributions addressing this topic are scattered throughout the literature (see, e.g., [18, 2, 1, 8, 12, 6, 11]). Normal ordering is predominantly studied in the context of univariate commutators, as the existence of a normal ordered form may be uncertain in the bivariate case (as discussed in <Ref>). Notable instances of expressions requiring normalization include binomials $(A+B)^n$ and powers of products $(AB)^n$. A more general encompassing instance was completely solved by Viskov [22]. This result was only recognized decades later by Mansour and Schork in [10], albeit focusing solely on the binomial case. Therefore, we aim to provide a comprehensive exposition of Viskov's result in <Ref> and demonstrate an application where the commutator $[B, A]$ is of the form $hA^s$ in <Ref>.

Before delving into Viskov's theorem, our objective is to introduce a formal exponentiation $A^B$ within noncommutative algebras, where both $A$ and $B$ are elements of the algebra. Utilizing Stirling numbers, we will establish its basic properties and explore its interaction with the exponential function (detailed in <Ref>). These new tools will enable us to establish the exponential identity under the constraint that $[B, A] = \alpha + \epsilon A - \lambda A^2$ (see <Ref>).

Additionally, in <Ref> we construct a new operator $\L$ that addresses a longstanding need, facilitating the transition between a normal ordered expression and its antinormal counterpart.

§ PRELIMINARIES

Let $\A$ be a associative $\K$-algebra (the field $\K$ is either $\R$ or $\C$). We'll assimilate elements of $\K$ being also as elements of $\A$ using the formula, for $x \in \K$, $x \defeq x \cdot \bm{1}_\A \in \A$, where $\bm{1}_\A$ is the neutral element according to the product of the algebra $\A$. It is not difficult to see that this notation is consistent with the use of power series, because for any power series $f$, $f(x) = f(x \cdot \bm{1}_\A) = f(x) \cdot \bm{1}_\A$.

When two elements $A$ and $B$ of $\A$ commute, i.e., $AB = BA$, we have the very known binomial theorem for $n \in \N$, the set of positive integers

\[
(A+B)^n = \sum_{k=0}^n \binom{n}{k} A^k B^{n-k}
\]

where $\binom{n}{k} \defeq \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

The binomial theorem directly implies the very important formula for the exponential function, defined as

\[
\exp A = e^A \defeq \sum_{n=0}^\infty \frac{A^n}{n!},
\]

which is the exponential identity

\[
e^{A+B} = e^A e^B.
\]

The goal of this paper is to give explicit expressions for the binomial theorem (BT for short) and exponential identity (EI) when $A$ and $B$ don't commute, depending on their commutator, denoted by Lie brackets

\[
[B, A] \defeq BA - AB.
\]

§.§ Stirling numbers

We will denote the falling factorial by the Pochhammer symbol $(x)_n \defeq x (x-1) \ldots (x-n+1)$ and define the Stirling numbers of the first kind $\coeff{n}{k}$ (and its signed version $s(n,k)$) by

\begin{equation}\label{e:stir1}
(x)_n = \sum_{k=0}^n \coeff{n}{k} (-1)^{n-k} x^k = \sum_{k=0}^n s(n,k) x^k,
\end{equation}

and of the Stirling number of the second kind $\Stir{n}{k}$ by <cit.>

\begin{equation}\label{e:stir2}
x^n = \sum_{k=0}^n \Stir{n}{k} (x)_k.
\end{equation}

Because theses numbers are defined as coefficients of an expression in some basis, if $k$ is an integer different from $0, \ldots, n$, then $\coeff{n}{k} = \Stir{n}{k} = 0$.
The well-known Stirling generating functions are

\begin{equation}\label{e:stir1gf}
\frac{(-\log(1-x))^k}{k!} = \sum_{n=k}^\infty \coeff{n}{k} \frac{x^n}{n!},
\end{equation}

\begin{equation}\label{e:stir2gf}
\frac{(e^x-1)^k}{k!} = \sum_{n=k}^\infty \Stir{n}{k} \frac{x^n}{n!}.
\end{equation}

Furthermore, the closed form formula for the Stirling numbers of the second kind is as follows

\begin{equation}\label{e:stirformula}
\Stir{n}{k} = \inv{k!} \sum_{i=0}^k \binom{k}{i} (-1)^{k-i} i^n.
\end{equation}

§.§ Exponentiation

By generalizing the binomial coefficient

\begin{equation}
\binom{x}{n} \defeq \frac{(x)_n}{n!},
\end{equation}

we can define, for $A, B \in \A$, the following formal power series

\begin{equation}\label{e:expo}
A^B \defeq \sum_{n=0}^\infty (A-1)^n \binom{B}{n}.
\end{equation}

Because we did not require the algebra to be commutative, in general, the form in which the power series is usually written is not true

\[
A^B \neq \sum_{n=0}^\infty \binom{B}{n} (A-1)^n.
\]

The convention we have adopted may not be ideal in all situations, but it offers the advantage of readability, ensuring that the reading order of $A$ and $B$ in $A^B$ aligns with the product order of its formula.

For most elements of most algebras, this series diverges (similar to divergence in $\C$). It should also diverge for $B = x$ when $x$ is negative and $A$ is not invertible in $\A$. However, any representation of exponentiation that does converge in $\A$ should satisfy <ref> as a valid formula when convergence occurs. Consequently, these representations should also exhibit the same properties as the one we will prove below.

For $A, B, C \in \A$  satisfying $[A, B] = [B, C] = 0$

* $A^B A^C = A^{B+C}$

* $(A^B)^C = A^{BC}$.

We are able to perform the Cauchy product because $A$ commutes with $B$ and Chu-Vandermonde's convolution because $B$ commutes with $C$

\[
A^B A^C = \sum_{n=0}^\infty (A-1)^n \sum_{k=0}^n \binom{B}{k} \binom{C}{n-k} = \sum_{n=0}^\infty (A-1)^n \binom{B+C}{n} = A^{B+C}.
\]

Now, it is easy to deduce that for positive integers $n$

\[
(A^B)^n = A^B A^B \cdots A^B = A^{B + B} \cdots A^B = \ldots = A^{nB},
\]

which we can use to fully prove the second point

\begin{align*}
&= \sum_{m=0}^\infty (A^B-1)^m \binom{C}{m} = \sum_{m=0}^\infty \sum_{k=0}^m \binom{m}{k} (-1)^{m-k} (A^B)^k \binom{C}{m} \\
&= \sum_{m=0}^\infty \sum_{k=0}^m \binom{m}{k} (-1)^{m-k} A^{kB} \binom{C}{m} \\
&= \sum_{m=0}^\infty \sum_{k=0}^m \binom{m}{k} (-1)^{m-k} \sum_{n=0}^\infty (A-1)^n \binom{kB}{n} \binom{C}{m},
\end{align*}

then we use <ref>

\begin{align*}
&= \sum_{n=0}^\infty \frac{(A-1)^n}{n!} \sum_{m=0}^\infty \sum_{k=0}^m \binom{m}{k} (-1)^{m-k}  \bra{\sum_{j=0}^n s(n,j) (kB)^j} \binom{C}{m} \\
&= \sum_{n=0}^\infty \frac{(A-1)^n}{n!} \sum_{m=0}^\infty  \sum_{j=0}^n s(n,j) B^j \binom{C}{m} \sum_{k=0}^m \binom{m}{k} (-1)^{m-k} k^j.
\end{align*}

In the inner sum, <ref> and (<ref>) lead to

\begin{align*}
\sum_{m=0}^\infty \frac{(C)_m}{m!} \sum_{k=0}^m \binom{m}{k} (-1)^{m-k} k^j = \sum_{n=0}^j (C)_m \Stir{j}{m} = C^j.
\end{align*}

The commutativity of $B$ and $C$ permits us to write $B^j C^j = (BC)^j$. Finally, applying <ref> once again, we arrive at

\[
(A^B)^C = \sum_{n=0}^\infty \frac{(A-1)^n}{n!} \sum_{j=0}^n s(n,j) B^j C^j = \sum_{n=0}^\infty \frac{(A-1)^n}{n!} (BC)_n = A^{BC}.
\]

The following proposition can be seen as a kind of notation trick, facilitated by our definition of exponentiation, to express this infinite sum more succinctly. It also enables us to utilize the properties we have just proven.

For $A, B\in\A$

\[
\sum_{n=0}^\infty \inv{n!} A^n B^n = (e^A)^B.
\]

We make use of <ref>, exchange the sums and conclude with <ref>

\begin{align*}
&= \sum_{k=0}^\infty (e^A - 1)^k \binom{B}{k} = \sum_{k=0}^\infty \sum_{n=k}^\infty \Stir{n}{k} \frac{A^n}{n!} (B)_k \\
&= \sum_{n=0}^\infty \frac{A^n}{n!} \sum_{k=0}^n \Stir{n}{k} (B)_k = \sum_{n=0}^\infty \frac{A^n}{n!} B^n.
\end{align*}

In the algebra of linear transformations over holomorphic functions, the most important operators are $\x$ and $D$, where $\x f(x) \defeq xf(x)$ and $D f(x) \defeq f'(x)$. Using <Ref> and Taylor's formula, we can express the generalization of the shift map $f(x) \mapsto f(x + g(x))$ in this manner

\begin{equation}\label{e:shift}
f(x + g(x)) = \sum_{n=0}^\infty \frac{g(x)^n}{n!} f^{(n)}(x) = \sum_{n=0}^\infty \frac{g(\x)^n}{n!} D^n f(x) = (e^{g(\x)})^D f(x),
\end{equation}

which appears to be new. <Ref> can also be seen as a composition operator, indeed

\begin{equation}
(e^{g(\x) - \x})^D f(x) = f(g(x)).
\end{equation}

§.§ Commutator

The commutator $[B, A] = BA - AB$ is a bilinear, anti-commutative operation that indicates the failure of commutativity between two elements of $\A$. These two properties combined allow for many manipulations that we will implicitly use throughout

\[
[A, B] = [A, A+B] = -[B, A] = [-B, A] = [B, -A].
\]

The commutator follows the simple identity

\begin{equation}\label{e:comid}
[B, A] = (A+B)^2 - A^2 - 2AB - B^2,
\end{equation}

due to the noncommutative expansion $(A+B)^2 = A^2 + AB + BA + B^2$.

We define normal ordering as the process of arranging a polynomial in terms of $A$ and $B$ such that all occurrences of $A$ appear to the left of products of $B$. If such an arrangement exists (we will see in <Ref> that it is not guaranteed), the normal ordered form is unique. When the $A$'s appear to the right of products with $B$, we'll refer to as antinormal ordering. Hence, what we refer to as a binomial theorem is simply the normal ordering of $(A+B)^n$. The normal ordered form of various other expressions has been studied in the literature [10]. Most notably, the normal ordered form of $(AB)^n$ is equally important as that of $(A+B)^n$ and will also be a subject of study.

The following lemma, reminiscent of the chain rule, is a classical result in normal ordering.

If $f$ is a power series and $[[B, A], A] = 0$ then

\[
[B, f(A)] = [B, A] f'(A).
\]

First we prove the result for monomials, and we see that the formula holds for $f(A) = 1$ and $f(A) = A$. By induction, we suppose for $n > 0$

\[
B A^n = A^n B + n [B, A] A^{n-1},
\]

now we multiply by $A$ to the right

\begin{align*}
B A^{n+1} &= A^n B A + n [B, A] A^n \\
&= A^n (AB + [B, A]) + n [B, A] A^n \\
&= A^{n+1} B + (n+1)[B, A]A^n,
\end{align*}

that is $[B, A^{n+1}] = (n+1)[B, A]A^n$. Due to the linearity of all operations, the extension to power series is easily obtained.

We define a derivation named the adjoint map by $\ad_B(A) \defeq [B, A]$. With this definition in place, we can express the analogue of the Leibniz rule below, along with its antinormal form.

If $A, B \in \A$ then for all integers $n$

\[
B^n A = \sum_{k=0}^n \binom{n}{k} \ad_B^k(A) B^{n-k} \quad\et\quad A B^n = \sum_{k=0}^n \binom{n}{k} B^{n-k} (-\ad_B)^k(A).
\]

The proofs are omitted as they can be readily accomplished by induction. We are now prepared to prove the main theorem of this section.

Let $p$ be a power series and $\A, \A'$ be two algebras respectively generated by $A, B$ and $C, D$, satisfying $[B, A] = p(A)$ and $[D, C] = p(D)$. Let $X \in \A$ uniquely written in normal-ordered form as

\[
X = \sum_{i, j} x_{i, j} A^i B^j.
\]

If we define the map $\L : \A \to \A'$ by

\[
\L(X) \defeq \sum_{i,j} x_{i,j} C^j D^i,
\]

* $\L$ is linear

* For $X, Y\in\A$, $\L(XY) = \L(Y) \L(X)$

* For a power series $f$, $\L(f(X)) = f(\L(X))$.

This theorem proves particularly valuable in the context of obtaining the antinormal form for normal-ordered expressions. We will illustrate this significance in <Ref>.

The linearity property follows from the unique decomposition into the normal ordered form. To establish the other property, we first express $Y$ in normal ordered form as $Y = \sum_{k, \ell} y_{k, \ell} A^k B^\ell$. Next we replicate this process for $XY$ using <Ref>:

\[
XY = \sum_{i,j,k,\ell} x_{i,j} y_{k,\ell} A^i B^j A^k B^\ell = \sum_{i,j,k,\ell} x_{i,j} y_{k,\ell} A^i \sum_{r=0}^j \binom{j}{r} \ad_B^r(A^k) B^{j-r} B^\ell.
\]

By repeatedly applying <Ref>, we deduce that $\ad_B^r(A^k)$ is a power series in $A$, confirming that the expression above is indeed a normal ordered form. With this understanding and as we want to apply $\L$, we now focus on elucidating the term $\L(A^i \ad_B^r(A^k) B^{j-r+\ell})$. Given that $[-C, D] = p(D)$ is analogous to $[B, A] = p(A)$, it's straightforward to see that this term simplifies to $C^{j-r+\ell} D^i \ad_{-C}^r(D^k)$. Thus, after applying $\L$, we arrive at

\[
\L(XY) = \sum_{i,j,k,\ell} x_{i,j} y_{k,\ell} \sum_{r=0}^j \binom{j}{r} C^{j-r+\ell} D^i \ad_{-C}^r(D^k).
\]

The above can be simplified using the simple fact that $\ad_{-C} = -\ad_C$ and the other formula of <Ref>

\begin{align*}
\L(XY)
&= \sum_{i,j,k,\ell} x_{i,j} y_{k,\ell} C^\ell \sum_{r=0}^j \binom{j}{r} C^{j-r}  (-\ad_C)^r(D^k) D^i \\
&= \sum_{i,j,k,\ell} x_{i,j} y_{k,\ell} C^\ell D^k C^j D^i = \bra{\sum_{k,\ell}  y_{k,\ell} C^\ell D^k} \bra{\sum_{i,j} x_{i,j} C^j D^i} \\
&= \L(Y) \L(X),
\end{align*}

as desired. By inducting on $n$, we then derive $\L(X^n) = \L(X)^n$, which readily extends to power series due to the linearity of $\L$.

In the very special case where the power series $p$ reduces to a constant $c$, we can simplify our choice for $C$ and $D$ by setting them to $A$ and $B$ respectively. The validity of the choice is ensured by $[B, A] = p(A) = p(B) = c$. We can opt for $A = \x$ and $B = D$ (where, here, $D$ is the derivative) since in this context, $c = 1$. This choice leads us to the intriguing formula for the composition operator of <Ref>

\[
\L((e^{g(\x)-\x})^D) = \sum_{n=0}^\infty \inv{n!} \L((g(\x) - \x)^n D^n) = \sum_{n=0}^\infty \inv{n!} \x^n (g(D) - D)^n = (e^\x)^{g(D) - D},
\]

which, according to <cit.>, is precisely an umbral operator when $g$ represents power series with a compositional inverse.

§.§ Baker-Campbell-Haussdorf formula

The Baker-Campbell-Haussdorf formula (BCH) is the solution for $Z$ of the equation $e^X e^Y = e^Z$ for non necessarily commuting $X, Y \in \A$, which takes the form of a (formal) Lie series, i.e., a series in iterated commutators of $X$ and $Y$. Later discovered, Dynkin's formula (BCHD) explicitly expresses the BCH. A clear exposition and proof of the BCHD are given in [15].

For $X, Y \in \A$,

\[
\log(e^X e^Y) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{\substack{m_1+n_1>0 \\ \cdots \\ m_k + n_k > 0}} \frac{\overbrace{[X,[\ldots, [X}^{m_1}, \overbrace{[Y, \ldots, [Y}^{n_1}, \overset{\substack{\dots\\[8pt]}}{[\ldots} \overbrace{[X, [\ldots, [X}^{m_k}, \overbrace{[Y, \ldots, [Y}^{n_k}, \ldots]\ldots]}{\bra{\sum_{i=1}^k m_i + n_i} \bra{\prod_{i=1}^k m_i! \cdot n_i!}},
\]

with the understanding that $[Y] \defeq Y$.

The reader should keep in mind that many terms in the expansion are zeros, for example, when the term $[Y, Y]$ appears.

The first few terms are

\begin{align*}
\log(e^X e^Y) &= X + Y + \inv2 [X, Y] + \inv{12} ([X, [X, Y]] + [Y [Y, X]]) - \inv{24} [Y, [X, [X, Y]]] \\
&- \inv{720} ([Y, [Y, [Y, [Y, X]]]] + [X, [X, [X, [X, Y]]]])\\
&+ \inv{360} ([X, [Y, [Y, [Y, X]]]] + [Y, [X, [X, [X, Y]]]]) \\
&+ \inv{120} ([Y, [X, [Y, [X, Y]]]] + [X, [Y, [X, [Y, X]]]]) \\
&+ \ldots
\end{align*}

The BCH aids in deriving the following important existence lemma for EIs.

$[B, A]$ is independent of $B$ if and only if there exists $\Psi_A : \K \to \A$ verifying $\Psi_A(0) = 1$ and $\Psi'_A(0) = A$, and such that for all $t \in \K$,

\[
e^{(A+B)t} = \Psi_A(t) e^{B t}.
\]

This lemma also assures the existence of the binomial theorem in this special case. Indeed, the formula can be retrieved after applications of the Cauchy product formula and identification of the coefficients of $t^n$.

( $\imp$ )
We know that

\[
[A+B, -B] = [B, A].
\]

So using <Ref>, we see that any combination of commutators of $A+B$ and $-B$ are functions of $A$.
Thus by the BCHD

\[
\log(e^{(A+B)t}e^{-B t}) = (A+B)t-Bt + \ldots = At + \ldots,
\]

where the dots are the commutator terms of higher order in $t$ that are independent of $B$ (see <Ref>).
Thus we can write $e^{(A+B)t} = e^{At + \ldots} e^{B t}$, and setting $\Psi_A(t) = e^{A t + \ldots}$, we notice that $\Psi_A$ has the properties we need.

( $\isimp$ )
Taking the second derivative at $t=0$ in the equality, we arrive at

\[
(A+B)^2 = \Psi_A''(0) + 2 \Psi_A'(0) B + \Psi_A(0)B^2 = \Psi_A''(0) + 2 A B + B^2,
\]

and by using <ref>, we recover $[B, A] = \Psi_A''(0) - A^2$, which is independent of $B$.

We can express the EI of <Ref> in various equivalent forms, which enables us to extend the formula we know (and prove in the following sections) to other similar cases.

\begin{align*}
[B, A] = f(A) &\implies e^{(A+B)t} = \Psi_A(t) e^{B t}
\shortintertext{\hspace{-29pt} then the following are also true}
\shortintertext{\item Symmetry}
[B, A] = f(B) &\implies e^{(A+B)t} = e^{A t} \Psi_B(t)
\shortintertext{\item Antinormal EI}
[B, A] = -f(A) &\implies e^{(A+B)t} = e^{Bt} \Psi_A(t)
\shortintertext{\item Special case of the BCH}
[B, A] = -f(A+B) &\implies e^{A t} e^{B t} = \Psi_{A+B}(t).
\end{align*}

For the symmetry, if $[B, A] = f(B)$ then $[-A-B, B] = f(B)$. Thus, according to the hypothesis

\[
e^{(B-A-B)t} = \Psi_B(t) e^{(-A-B)t}.
\]

In other words,

\[
e^{(A+B)t} = e^{A t} \Psi_B(t).
\]

The special case of the BCH is treated similarly by noticing that $[B, A] = -f(A+B)$ is the same as $[-B, A+B] = f(A+B)$.
For the antinormal EI, we could just use the symmetry with $[A, B] = f(A)$. Alternatively, we can use the operator $\L$ of <Ref> which was designed for this purpose, and make the following computation

\begin{align*}
\L(e^{(A+B)t}) &= \L(\Psi_A(t) e^{Bt}) \\
e^{\L(A+B)t} &= \L(e^{Bt}) \L(\Psi_A(t)) \\
e^{(\L(A)+\L(B))t} &= e^{\L(B)t} \Psi_{\L(A)}(t) \\
e^{(C+D)t} &= e^{Dt} \Psi_C(t),
\end{align*}

with $[C, D] = -f(D)$ and $C$ and $D$ that we can freely rename to $A$ and $B$.

The following proposition is an additional general tool, for computing EIs, generalizing <Ref>, that will prove useful later on.

If $[B, A]$ doesn't depend on $B$ then for $\Psi_A$ defined in <Ref>, and for all $t, \lambda \in \K$, we have the following

\[
e^{(\lambda A + B)t} = \Psi_A(t)^\lambda e^{B t}.
\]

We first prove the proposition for $\lambda = n \in \N$. Because $[(n-1)A + B, A] = [B,A]$ doesn't depend on $B$, we have

\[
e^{(nA+B)t} = e^{(A+(n-1)A+B)t} = \Psi_A(t) e^{((n-1)A+B)t}.
\]

Repeating the process another $n-1$ times, we recover the expected term $\Psi_A(t)^n e^{B t}$.

Now to generalize this to any $\lambda$, we make a calculation similar to the proof of <Ref>, where we first use the definition of exponentiation and the result above

\begin{align*}
\Psi_A(t)^\lambda e^{B t}
&= \sum_{k=0}^\infty \binom{\lambda}{k} \sum_{j=0}^k \binom{k}{j} (-1)^{k-j} \Psi_A(t)^j e^{B t} \\
&= \sum_{k=0}^\infty \binom{\lambda}{k} \sum_{j=0}^k \binom{k}{j} (-1)^{k-j} e^{(jA + B) t}
\end{align*}

We'll quickly introduce the notation $\uline{A^i B^j}$ which represents the sum of all possible product orders for $i$ $A$'s and $j$ $B$'s. For example, $\uline{A^2 B} = \uline{ABA} = \uline{BA^2} \defeq A^2 B + ABA + BA^2$ and the notation carries this property: $\uline{(\lambda A)^i (\mu B)^j} = \lambda^i \mu^j \uline{A^i B^j}$. We can now write the noncommutative expansion of a binomial as

\[
(A+B)^n = \sum_{k=0}^n \uline{A^k B^{n-k}},
\]

\begin{align*}
\Psi_A(t)^\lambda e^{B t}
&= \sum_{k=0}^\infty \binom{\lambda}{k} \sum_{j=0}^k \binom{k}{j} (-1)^{k-j}  \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{i=0}^n \uline{(jA)^i B^{n-i}} \\
&= \sum_{n=0}^\infty  \frac{t^n}{n!} \sum_{i=0}^n \sum_{k=0}^\infty \binom{\lambda}{k} \bra{\sum_{j=0}^k \binom{k}{j}(-1)^{k-j} j^i} \uline{A^i B^{n-i}},
\end{align*}

using the closed form of the Stirling numbers of the second kind (<ref>) and then their definition (<ref>), we conclude the computation

\begin{align*}
\Psi_A(t)^\lambda e^{B t}
&= \sum_{n=0}^\infty  \frac{t^n}{n!} \sum_{i=0}^n \bra{\sum_{k=0}^\infty \binom{\lambda}{k}  \Stir{i}{k} k!} \uline{A^i B^{n-i}} \\
&= \sum_{n=0}^\infty  \frac{t^n}{n!} \sum_{i=0}^n \bra{ \sum_{k=0}^i \Stir{i}{k} (\lambda)_k} \uline{A^i B^{n-i}} \\
&= \sum_{n=0}^\infty  \frac{t^n}{n!}  \sum_{i=0}^n \lambda^i \uline{A^i B^{n-i}} = \sum_{n=0}^\infty  \frac{t^n}{n!}  (\lambda A + B)^n \\
&= e^{(\lambda A + B)t}.
\end{align*}

§ EXPONENTIAL IDENTITY AND BINOMIAL THEOREM WITH QUADRATIC COMMUTATOR

In the subsequent sections, we will establish the validity of EIs under the condition

\[
[B, A] = \alpha + \epsilon A - \lambda A^2,
\]

that is when the commutator is at most quadratic. We denote the algebra generated by $A$ and $B$ with $\A(\alpha, \epsilon, -\lambda)$.

This analysis readily extends to cases where the polynomial is in $B$, using <Ref>. While the most general formula presents a complex formulation, making an exhaustive derivation of its corresponding BT impractical, there exist numerous limiting cases where significant simplifications occur, often accompanied by concise BTs and where the generated algebra is well-known. The cases we'll study are as follows

* The most general quadratic case has, therefore, no straightforward binomial (<Ref>).

* $\lambda = 1$ (<Ref>) which appears to be the only value for $\lambda$ that yields a simple binomial.

* $\epsilon = 0$ (<Ref>) which remains too complex to carry a nice binomial.

* $\alpha = 0$  (<Ref>), which surprisingly does have a binomial.

* $\lambda = 0$ (<Ref>). $\A(\alpha, \epsilon, 0)$ is referred to as the generalized Ore algebra [11] and lacks a binomial.

* $\alpha = \epsilon = 0$ (<Ref>). $\A(0, 0, -\lambda)$ is the well-known Jordan plane, also sometimes called the meromorphic Weyl algebra [14], with a trivial scaling.

* $\alpha = \lambda = 0$ (<Ref>). $\A(0, \epsilon, 0)$ is the well-known (scaled) shift algebra.

* $\epsilon = \lambda = 0$ (<Ref>). $\A(\alpha, 0, 0)$ is the well-known (scaled) Weyl algebra.

To express the first two results more clearly, we introduce, for $A \in \mathcal{A}$, a power series in $t$, continuous in all variables $t, x, y \in \mathbb{K}$:

\begin{equation}\label{e:Phi}
\Phi_A(t, x, y) \defeq
\begin{cases}
\dfrac{y e^{x t} - x e^{y t}}{y - x} + A \dfrac{e^{y t} - e^{x t}}{y - x} & \com{if} x \neq y\\[8pt]
(1-xt)e^{x t} +  A t e^{x t} & \com{if} x = y,
\end{cases}
\end{equation}

furthermore, it will prove useful to define

\[
[x]_n \defeq \prod_{i=1}^{n-1} (1 - i x),
\]

which can be alternatively expressed, for $x \neq 0$, as $[x]_n = x^n (1/x)_n$.

§.§ The preliminary case lambda=1

We will begin our investigation with what may appear as an arbitrary case: when $\lambda = 1$ in the equation $[B, A] = \alpha + \epsilon A - \lambda A^2$ where neither $\alpha$ nor $\epsilon$ are zero at the same time (subject of <Ref>). Remarkably, $1$ seems to be the only value for $\lambda$ for which the formulation of the BT in <Ref> is simple. This observation stems from the simplification of the squared term in $(A+B)A = A^2 + AB + \alpha + \epsilon A - A^2 = AB + \alpha + \epsilon A$ (as we'll witness in the proof). Furthermore,  the study of this special case constitutes a necessary initial step towards addressing the more general case of <Ref>.

Let $A$ and $B$ be two elements of $\A$, $\alpha$ and $\epsilon$ two complex numbers, such that $\alpha \neq 0$ or $\epsilon \neq 0$. Let $r, \rho$ be the two roots of the polynomial $\alpha + \epsilon X - X^2$.

Let $(u_n)_{n\in\N}, (v_n)_{n\in\N}$ two complex sequences defined by

\[
\begin{cases}
u_0 = 0,\, u_1 = 1 \\
u_{n+2} = \epsilon u_{n+1} + \alpha u_n
\end{cases}
\]

\[
\begin{cases}
v_0 = 1,\, v_1 = 0 \\
v_{n+2} = \epsilon v_{n+1} + \alpha v_n
\end{cases}
\]

or equivalently by

\[
u_n = \begin{cases}
\dfrac{\rho^n - r^n}{\rho-r} & \com{if} r \neq \rho \\[8pt]
n r^{n-1} & \com{if} r = \rho
\end{cases}
\]

\[
v_n = \begin{cases}
\dfrac{\rho r^n - r\rho^n}{\rho-r} & \com{if} r \neq \rho \\[8pt]
(1-n)r^n & \com{if} r = \rho.
\end{cases}
\]

Then these statements are equivalent

* []$[B, A] = \alpha + \epsilon A - A^2$

* for $n\in\N$,
\[
(A+B)^n = \sum_{k=0}^n \binom{n}{k} \bra{v_k  + u_k A} B^{n-k}
\]

* for $t\in\K$,
\[
e^{(A+B)t} = \Phi_A(t, r, \rho) e^{B t}.
\]

We notice this elegant relation where a single equation holds equivalence to a countably infinite number of equations, which in turn is also equivalent to an uncountably infinite number of equations.

The relationship between the coefficients and roots of a polynomial is $ \alpha = - r \rho$ and $\epsilon = r + \rho$. We can employ these alongside the explicit expressions of $u_n$ and $v_n$ to derive the relationship

\[
v_{n+1} = \alpha u_n \quad \et \quad u_{n+1} = v_n + \epsilon u_n. \tag{$*$}
\]

We proceed to prove by induction (i $\imp$ ii). Since the base case $n=0$ is satisfied, let us assume the property holds for $n \in \mathbb{N}$.

\begin{align*}
& (A+B)^{n+1} = \sum_{k=0}^n \binom{n}{k} (A+B)\bra{v_k + u_k A} B^{n-k} \\
={}& \sum_{k=0}^n \binom{n}{k} \bra{v_k A + u_k A^2 + v_k B + u_k (AB+ \alpha + \epsilon A - A^2)} B^{n-k} \\
={}& \sum_{k=0}^n \binom{n}{k} (u_k AB +  v_k B + (v_k + \epsilon u_k)A + \alpha u_k) B^{n-k},
\end{align*}

where we split the sum and apply $(*)$ to obtain

\begin{align*}
&= \sum_{k=0}^n \binom{n}{k} (u_k A +  v_k) B^{n+1-k} + \sum_{k=0}^n \binom{n}{k} (u_{k+1}A + v_{k+1} B) B^{n-k} \\
&= \sum_{k=0}^{n+1} \bra{\binom{n}{k} + \binom{n}{k-1}} \bra{v_k + u_k A} B^{n+1-k},
\end{align*}

and finally, utilizing Pascal's identity, we conclude the induction.

For the proof of (ii $\imp$ i), first we compute

\[
(A+B)^2 = (v_0 + u_0 A) B^2 + 2 (v_1 + u_1A)B + v_2 + u_2A = B^2 + 2 AB + \alpha + \epsilon A,
\]

then we make use of <ref> to obtain $[B, A] = \alpha + \epsilon A -  A^2$, as expected.

For (iii $\imp$ ii), we notice that by the explicit formula for $(u_n)_{n\in\N}$ and $(v_n)_{n\in\N}$, we precisely have

\[
\Phi_A(t, r, \rho) =  \sum_{n=0}^\infty \frac{t^n}{n!} (v_n + u_n A) ,
\]

which is a Cauchy product with $e^{Bt}$ and identification away from the desired result. Similarly, we derive (ii $\imp$ iii) by summing the binomial theorem with the factor $t^n/n!$.

§.§ The general case and its various limiting cases

<Ref> was a mandatory step in proving the result in full generality, as we were able to connect the commutator equation i with the EI iii through the BT ii. This would have been considerably more challenging to achieve in the general case, given the absence of straightforward formulations for the BT.

Let $A$ and $B$ be two elements of $\A$ and $\alpha$, $\epsilon$ and $\lambda \neq 0$ three complex numbers, such that $\alpha \neq 0$ or $\epsilon \neq 0$.
Let $r, \rho$ be two roots of the polynomial $\alpha + \epsilon X - \lambda X^2$.
Then these statements are equivalent

* []$[B, A] = \alpha + \epsilon A - \lambda A^2$

* for $t\in\K$,
\[
e^{(A+B)t} = \Phi_A(\lambda t, r, \rho)^{1/\lambda} e^{B t}.
\]

It's worth mentioning that a fairly simple BT could be derived, but this wouldn't come without the radicals of the roots.

According to <Ref>, we are aware of the existence of $\Psi_A$ such that

\begin{equation}\label{e:psilambda}
e^{(\lambda A + B) t} = \Psi_A(t)^\lambda e^{B t}.
\end{equation}

Moreover, equivalently to i, we find, through linearity, that $[B, \lambda A] = \lambda \alpha + \epsilon (\lambda A) - (\lambda A)^2$, thus according to <Ref>

\[
e^{(\lambda A + B) t} = \Phi_{\lambda A}(t, \lambda r, \lambda\rho) e^{B t},
\]

because $\lambda r, \lambda \rho$ are roots of the polynomial $\lambda \alpha + \epsilon X - X^2$ . We can identify the above expression with <ref> and simplify it via the definition of $\Phi$, to obtain

\[
\Psi_A(t)^\lambda = \Phi_A(\lambda t, r, \rho),
\]

and by raising this equation to the power $1/\lambda$ (with <Ref>), we arrive at the desired result.

The proof is highly specific to the degree 2 case, and it does not appear that it can be generalized to the case of the commutator being a polynomial of higher degree. This limitation likely arises because the degree 2 cases are more natural and straightforward, while higher-degree cases may introduce additional complexities that are not easily addressed within the framework of the proof above.

Let $A$ and $B$ be two elements of $\A$, $\alpha \neq 0$ and $\lambda \neq 0$ two complex numbers.
Then these statements are equivalent

* []$[B, A] = \alpha - \lambda A^2$

* for $t\in\K$,
\[
e^{(A+B)t} = \bra{\cosh(\sqrt{\lambda \alpha}t) + A \sqrt{\frac{\lambda}{\alpha}} \sinh(\sqrt{\lambda\alpha}t)}^{1/\lambda} e^{B t}.
\]

Using the identity $(\cosh x + A \sinh x)^p = (1 - (\tanh x)^2)^{-p/2} (1 + A \tanh(x))^p$, one can derive a BT for this specific instance however, they would include coefficients of powers of the hyperbolic tangent that are not easily expressible.

Looking at <Ref>, with $\epsilon = 0$ we have $r, \rho = \pm \sqrt{\alpha/\lambda}$. The proof is then straightforward.

Let $A$ and $B$ be two elements of $\A$, $\lambda \neq 0$ and $\epsilon \neq 0$ two complex numbers.
Then these statements are equivalent

* []$[B, A] = \epsilon A - \lambda A^2$

* for $n\in\N$,
\begin{align*}
(A+B)^n &= \sum_{k=0}^n \binom{n}{k} \bra{\sum_{j=0}^k \Stir{k}{j} \epsilon^{k-j} [\lambda]_j A^j} B^{n-k} \\
&= \sum_{k=0}^n \epsilon^{n-k} \sum_{j=0}^k  \binom{n}{j} \Stir{n-j}{k-j} [\lambda]_{k-j} A^{k-j} B^j
\end{align*}

* for $t\in\K$,
\[
e^{(A+B)t} = \bra{1+\lambda A \frac{e^{\epsilon t}-1}{\epsilon}}^{1/\lambda} e^{B t}.
\]

Considering <Ref>, with $\alpha = 0$, we find $r=0$ and $\rho = \epsilon/\lambda$. The proof of (i $\eq$ iii) is straightforward. For ii, we utilize the EI. First, we express the following in its power series form and then use the Stirling generating function (<ref>)

\begin{align*}
&\bra{1+\lambda A \frac{e^{\epsilon t}-1}{\epsilon}}^{1/\lambda}
= \sum_{j=0}^\infty \binom{1/\lambda}{j} \lambda^j A^j \bra{\frac{e^{\epsilon t}-1}{\epsilon}}^j \\
={}& \sum_{j=0}^\infty (1/\lambda)_j \lambda^j A^j \inv{\epsilon^j }\sum_{n=j}^\infty \Stir{n}{j} \frac{(\epsilon t)^n}{n!} = \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{j=0}^n \Stir{n}{j} \epsilon^{n-j} [\lambda]_j A^j.
\end{align*}

By the Cauchy product and identification

\[
(A+B)^n = \sum_{k=0}^n \binom{n}{k} \bra{\sum_{j=0}^k \Stir{k}{j} \epsilon^{k-j} [\lambda]_j A^j} B^{n-k}.
\]

The second formula is obtained by summing the triangular array of terms in the orthogonal direction

\begin{equation}\label{e:sumarray}
\sum_{0\leq j\leq k\leq n} c_{j,k} = \sum_{0\leq j\leq k\leq n} c_{k-j,n-j}.
\end{equation}

Let $A$ and $B$ be two elements of $\A$, $\epsilon \neq 0$ and $\alpha$ two complex numbers.
Then these statements are equivalent

* []$[B, A] = \alpha + \epsilon A$

* for $t\in\K$,
\[
e^{(A+B)t} = e^{\tfrac{\alpha}{\epsilon}\bra{\tfrac{e^{\epsilon t}-1}{\epsilon} - t}} e^{A \tfrac{e^{\epsilon t}-1}{\epsilon}} e^{B t}
\]

Beginning with <Ref> and aiming to take the limit $\lambda \to 0$, we select $|\lambda| > 0$ sufficiently small such that the discriminant $\Delta^2 \defeq \epsilon^2 + 4 \lambda\alpha$ remains non-zero. The distinct roots can then be expressed as

\[
r = \frac{\epsilon-\Delta}{2\lambda} = -\frac{\alpha}{\epsilon} + o(1), \quad \rho = \frac{\epsilon+\Delta}{2\lambda} = \frac{\epsilon}{\lambda} + o(\lambda^{-1}),
\]

because $\Delta = \epsilon + 2 \lambda\alpha/\epsilon + o(\lambda)$. Utilizing $\rho - r = \Delta/\lambda$, we then obtain

\begin{align*}
\log(\Phi_A(\lambda t, r, \rho)^{1/\lambda})
&= \inv\lambda (\Phi_A(\lambda t, r, \rho) - 1) + o(1) \\
&= \inv{\Delta} (\rho e^{r \lambda t} - r e^{\rho \lambda t} + A (e^{\rho \lambda t} - e^{r \lambda t}) - (\rho - r)) + o(1) \\
&= \inv{\Delta} (\rho (e^{r \lambda t} - 1) - r (e^{\rho \lambda t} - 1) + A (e^{\rho \lambda t} - e^{r \lambda t})) + o(1) \\
&\xrightarrow[\lambda\to0]{} \inv\epsilon \bra{-\alpha t + \frac{\alpha}{\epsilon}(e^{\epsilon t} - 1) + A (e^{\epsilon t} - 1)},
\end{align*}

which upon applying the exponential yields the EI of the corollary.

A notable implication of the theorem is a specific instance of the BCH. By using <Ref>, we can make the computation

\[
e^{\bra{\tfrac{\epsilon t}{e^{\epsilon t}-1}A + B}t} = \bra{e^{\tfrac{\alpha}{\epsilon}\bra{\tfrac{e^{\epsilon t}-1}{\epsilon} - t}} e^{A \tfrac{e^{\epsilon t}-1}{\epsilon}}}^{\tfrac{\epsilon t}{e^{\epsilon t}-1}} e^{B t} = e^{\tfrac{\alpha t}{\epsilon}\bra{1 - \tfrac{\epsilon t}{e^{\epsilon t}-1}}} e^{A t} e^{B t},
\]

which can also be expressed as

\begin{equation}
e^{At} e^{Bt} = e^{\bra{A+B + [B, A] \bra{\tfrac{t}{e^{\epsilon t}-1}-\tinv{\epsilon}}}t}.
\end{equation}

This formula is a special case of the bivariate $[B, A] = \alpha + \epsilon A + \sigma B$, for which a closed form of the BCH was established in [19].

In the rest of this section, the results we present are already known. For example, in this instance, known as the meromorphic Weyl algebra, the EI is commonly referred to as the "Berry identity" <cit.>. Viskov [23] first expanded it into the BT, but it was also notably independently found by Benaoum [1], along with the $q$-analogue.

Let $A$ and $B$ be two elements of $\A$, $\lambda \neq 0$ a complex number.
Then these statements are equivalent

* []$[B, A] = -\lambda A^2$

* for $n\in\N$,
\[
(A+B)^n = \sum_{k=0}^n  \binom{n}{k} [\lambda]_k A^k B^{n-k}
\]

* for $t\in\K$,
\[
e^{(A+B)t} = \bra{1+\lambda At }^{1/\lambda} e^{B t}.
\]

Using <Ref> with  $\epsilon \to 0$ and the limit

\[
\lim_{\epsilon \to 0} \frac{e^{\epsilon t}-1}{\epsilon} = t.
\]

The EI in the shift algebra is known as Sack's identity [18] and Viskov [21] later expressed the BT by expansion of Sack's identity.

Let $A$ and $B$ be two elements of $\A$, $\epsilon \neq 0$ a complex number.
Then these statements are equivalent

* []$[B, A] = \epsilon A$

* for $n\in\N$,
\begin{align*}
(A+B)^n &= \sum_{k=0}^n \binom{n}{k} \bra{\sum_{j=0}^k \Stir{k}{j} \epsilon^{k-j} A^j} B^{n-k} \\
&= \sum_{k=0}^n \epsilon^{n-k} \sum_{j=0}^k  \binom{n}{j} \Stir{n-j}{k-j} A^{k-j} B^j
\end{align*}

* for $t\in\K$,
\[
e^{(A+B)t} = e^{A \tfrac{e^{\epsilon t}-1}{\epsilon}} e^{B t}.
\]

Using <Ref> for ii with $\lambda = 0$ and <Ref> for iii with $\alpha = 0$.

The subsequent result is arguably one of the most renowned, as it encompasses the "Glauber formula", which is a well-known and useful result in quantum mechanics, as the underlying mathematical structure is often the Weyl algebra. Although the binomial theorem is a direct consequence of the Glauber formula, it seems that it was first formulated by Yamazaki [24].

Let $A$ and $B$ be two elements of $\A$, $\alpha$ a complex number.
Then these statements are equivalent

* []$[B, A] = \alpha$

* for $n\in\N$,
\[
(A+B)^n = \sum_{i=0}^{\floor{n/2}} \frac{n! (\alpha/2)^i}{i!(n-2i)!} \sum_{k=0}^{n-2i} \binom{n-2i}{k} A^k B^{n-2i-k}
\]

* for $t\in\K$,
\[
e^{(A+B)t} = e^{\alpha \tfrac{t^2}{2}} e^{A t} e^{B t}.
\]

Using <Ref> for iii with $\epsilon \to 0$ and the limits

\[
\lim_{\epsilon \to 0} \frac{e^{\epsilon t}-1}{\epsilon} = t, \qquad \lim_{\epsilon \to 0} \inv{\epsilon} \bra{\frac{e^{\epsilon t}-1}{\epsilon}-t} = \frac{t^2}{2}.
\]

The BT follows from expanding the EI. We apply the Cauchy product to the following power series

\[
e^{\alpha \tfrac{t^2}{2}} = \sum_{p=0}^\infty t^p \frac{(\alpha/2)^{p/2}}{(p/2)!} [p \text{ is even}]
\]

\[
e^{A t} e^{B t} = \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{k=0}^n \binom{n}{k} A^k B^{n-k},
\]

(where we have used Iverson's brackets) to obtain

\begin{align*}
e^{\alpha \tfrac{t^2}{2}} e^{A t} e^{B t}
&= \sum_{n=0}^\infty t^n \sum_{j=0}^n \frac{(\alpha/2)^{j/2}}{(j/2)!(n-j)!} [j \text{ is even}] \sum_{k=0}^{n-j} \binom{n-j}{k} A^k B^{n-j-k} \\
&= \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{i=0}^{\floor{n/2}} \frac{n! (\alpha/2)^i}{i!(n-2i)!} \sum_{k=0}^{n-2i} \binom{n-2i}{k} A^k B^{n-2i-k}.
\end{align*}

Hence by identification with $e^{(A+B)t}$, we prove ii.

§ THE CASE OF A BIVARIATE COMMUTATOR

The existence of an EI and a BT is guaranteed when the commutator is univariate, as established in <Ref> and <Ref>.i.  However, this guarantee is lost in the bivariate case, as noted in [17]. Nonetheless, as hinted at by <Ref>.iii, the BCH formula and its less renowned counterpart, the Zassenhaus formula[which, in short, provides an expression for $e^{-A}e^{-B}e^{A+B}$], appear to be more promising paths for investigating bivariate commutators. Notably, closed forms for both the BCH [19] and the Zassenhaus formula [6] have been derived in the case $[A, B] = \alpha + \epsilon A + \sigma B$.

Additional formulas of the BCH in bivariate settings can be derived using <Ref>.iii and the results from <Ref>. One such formula is presented in the following theorem.

Let $A$ and $B$ be two elements of $\A$, $\epsilon, \lambda \neq 0$ two complex numbers.
Then these statements are equivalent

* []$[B, A] = \epsilon (A+B) + \lambda (A+B)^2$

* for $n\in\N$,
\[
[\lambda]_n (A+B)^n = \sum_{k=0}^n \coeff{n}{k}  \epsilon^{n-k} \sum_{j=0}^k \binom{k}{j} A^j B^{k-j} \\
\]

* for $t\in\K$,
\[
e^{At} e^{B t} = \bra{1+\lambda (A+B) \frac{1-e^{-\epsilon t}}{\epsilon}}^{1/\lambda}.
\]

iii follows from the exponential identity of <Ref> used in the form of the special case of the BCH given by <Ref>. For ii, we first rewrite iii as

\[
e^{A \tfrac{-\log(1-\epsilon t)}{\epsilon}} e^{B \tfrac{-\log(1-\epsilon t)}{\epsilon}} = (1+\lambda (A+B)t)^{1/\lambda}.
\]

In the left-hand side, we make use of <ref> after applying the Cauchy product

\begin{align*}
&e^{A \tfrac{-\log(1-\epsilon t)}{\epsilon}} e^{B \tfrac{-\log(1-\epsilon t)}{\epsilon}} = \sum_{k=0}^\infty \frac{(-\log(1-\epsilon t))^k}{\epsilon^k k!} \sum_{j=0}^k \binom{k}{j} A^j B^{k-j} \\
={}& \sum_{k=0}^\infty \sum_{n=k}^\infty \coeff{n}{k} \frac{t^n}{n!} \epsilon^{n-k} \sum_{j=0}^k \binom{k}{j} A^j B^{k-j} = \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{k=0}^n \coeff{n}{k}  \epsilon^{n-k} \sum_{j=0}^k \binom{k}{j} A^j B^{k-j},
\end{align*}

and the right-hand side becomes, after a calculation done earlier

\[
(1+\lambda (A+B)t)^{1/\lambda} = \sum_{n=0}^\infty \frac{t^n}{n!} (A+B)^n [\lambda]_n,
\]

which means that by identification, we finally obtain

\[
[\lambda]_n (A+B)^n = \sum_{k=0}^n \coeff{n}{k}  \epsilon^{n-k} \sum_{j=0}^k \binom{k}{j} A^j B^{k-j}.
\]

Here, we observe an interesting phenomenon that may appear contradictory: when $[B, A]$ contains degree two terms in both $A$ and $B$, one may encounter a binomial theorem with rational coefficients (after dividing by $[\lambda]_n$ in ii), even when all parameters are integers.

To gain a better understanding of this occurrence, we aim to illustrate a special case proposed in [17], namely, the case $[B, A] = \lambda A^2 + \mu B^2$. The reader may wish to attempt on their own to normal order the term $B^2 A$, and might stop when arriving at

\[
B^2 A = 2 \lambda^2 A^3 + 2\lambda A^2 B + (1+\lambda\mu)A B^2 + 2 \mu B^3 + \lambda \mu B^2 A,
\]

because of the "recursive" aspect in $B^2 A$ of the formula. Thus, as noticed in [17], the normal ordered form of $B^2 A$ exists if and only if $\lambda \mu \neq 1$, and it is the following.

\[
B^2 A = \frac{2 \lambda^2}{1-\lambda\mu} A^3 + \frac{2\lambda}{1-\lambda\mu} A^2 B + \frac{1+\lambda\mu}{1-\lambda\mu}A B^2 + \frac{2 \mu}{1-\lambda\mu} B^3.
\]

Note that the analogue of "$\lambda\mu \neq 1$" in the case of <Ref> is "$\lambda$ is not a root of $[X]_n$". Therefore, in the limiting case $\lambda\to0$, the binomial always exists and is given below.

Let $A$ and $B$ be two elements of $\A$, $\epsilon \neq 0$ a complex number.
Then these statements are equivalent

* []$[B, A] = \epsilon (A+B)$

* for $n\in\N$,
\[
(A+B)^n = \sum_{k=0}^n \coeff{n}{k}  \epsilon^{n-k} \sum_{j=0}^k \binom{k}{j} A^j B^{k-j} \\
\]

* for $t\in\K$,
\[
e^{At} e^{B t} = e^{(A+B) \tfrac{1-e^{-\epsilon t}}{\epsilon}}.
\]

In this scenario, $A$ and $B$ would serve as generators of the (scaled) excedance algebra [4]. Additionally, alongside <Ref>, <Ref> represents another special case of the result presented in [19]. This case can be readily transformed into an EI or even a Zassenhaus formula, which would then constitute a special case of [6].

§ A THEOREM FROM VISKOV

In <cit.>, a special case of a theorem from Viskov <cit.> is briefly mentioned. However, it appears that the broader applicability of the more general theorem was not recognized within the context of the theory being discussed. Therefore, we aim to present a more general version of this theorem by Viskov. This theorem is remarkably powerful, to the extent that it implies <Ref> and the theorems outlined in <Ref>.

Regarding the history of this theorem, Viskov initially formulated a less general but multidimensional result in [20]. Subsequently, in [23], Viskov extended the uni-dimensional case of his result and utilized a special instance of it to reaffirm the validity of Sack's identity [21]. A further generalization was presented in [22][Although [21, 22] were published before [23], the manuscript of the latter was received prior to the former.]. Below, we present the version from [23], as the additional generality from [22] isn't pertinent.

It's worth noting that Viskov's proof in [22], which broadly follows the outline provided in [23], contains an error that we intend to rectify.

Let $p, f, g$ be formal power series and $A, B \in \A$ such that $[B, A] = p(A)$. Then

\[
\exp((f(A) B + g(A))t) = e^\gamma (e^\phi)^B,
\]

\[
\phi = \int_0^t f(\alpha(\tau)) \odif\tau, \qquad \gamma = \int_0^t g(\alpha(\tau)) \odif\tau,
\]

where $\alpha$ is the solution to the Cauchy problem

\[
\odv{\alpha}{t} = p(\alpha) f(\alpha), \quad \alpha(0) = A.
\]

First, we will prove the special case, $p = 1$ and then retrieve the full theorem by substitution. With $[B, A] = p(A) = 1$, we choose the representatives $B = D$ and $A = \x$. For a polynomial $q$, we consider the expression

\[
u = u(t, x) \defeq \exp\bra{\bra{f(\x)D + g(\x)}t} q(x).
\]

$u$ is obviously the unique solution to the Cauchy problem

\begin{equation}\label{e:pde}
\pdv{u}{t} = f(x) \pdv{u}{x} + g(x) u, \quad u(0, x) = q(x).
\end{equation}

We will show that $v \defeq e^\gamma q(x+\phi)$ satisfy the above, which will yield, by unicity, equality with $u$. As in the theorem, $\phi$ and $\gamma$ are defined by

\begin{equation}\label{e:phi}
\phi(t, x) \defeq \int_0^t f(\alpha(\tau, x)) \odif\tau,
\end{equation}

\begin{equation}\label{e:gamma}
\gamma(t, x) \defeq \int_0^t g(\alpha(\tau, x)) \odif\tau,
\end{equation}

with $\alpha$ being the unique solution to this Cauchy problem

\begin{equation}\label{e:alpha}
\pdv{\alpha}{t} = f(\alpha), \quad \alpha(0, x) = x.
\end{equation}

Immediately with (<ref>) and (<ref>), we have

\begin{equation}\label{e:mu}
\phi(t, x) = \int_0^t \pdv{\alpha}{t}(\tau, x) \odif\tau = \alpha(t, x) - x.
\end{equation}

This simplifies $v$ to $e^\gamma q(\alpha)$. Now we'll derive the partial differential equation satisfied by $\gamma$ and $\alpha$. <Ref> implies

\begin{equation}\label{e:t}
t = \int_0^t \pdv{\alpha}{t} \frac{\odif\tau}{f(\alpha)},
\end{equation}

and in <ref>, we make the substitution $z = \alpha$. The differential is given by $\odif z = \pdv{\alpha}{\tau} \odif\tau = f(\alpha) \odif\tau = f(z) \odif\tau$, which respectively yields

\[
t = \int_x^{\alpha(t, x)} \frac{\odif z}{f(z)},
\]

\[
\gamma(t, x) = \int_x^{\alpha(x, t)} \frac{g(z)}{f(z)} \odif z.
\]

Now we differentiate with respect to $x$

\[
0 = \pdv{\alpha}{x} \inv{f(\alpha)} - \inv{f(x)}, \tag{$*$}
\]

which we rewrite with <ref> as

\[
\pdv{\alpha}{t} = f(x) \pdv{\alpha}{x}.
\]

\[
\pdv{\gamma}{x} = \pdv{\alpha}{x} \frac{g(\alpha)}{f(\alpha)} - \frac{g(x)}{f(x)},
\]

then we use <ref> and ($*$) to obtain

\[
\pdv{\gamma}{t} = f(x) \pdv{\gamma}{x} + g(x).
\]

These equations suffice to show that $v = e^\gamma q(\alpha)$ does satisfy <ref>. Therefore, by the unicity of the solution to the Cauchy problem, we have

\[
\exp((f(\x)u + g(\x))t) q(x) = e^{\gamma(t, x)} q(x+\phi(t, x)) = e^{\gamma(t, \x)} (e^{\phi(t, \x)})^D q(x),
\]

where we have used the notation of <Ref>. The above relation holds for all polynomials $q$, hence the equality of the operators

\begin{equation}\label{e:opeq}
\exp((f(\x)u + g(\x))t) =  e^{\gamma(t, \x)} (e^{\phi(t, \x)})^D.
\end{equation}

where we have rewritten the right hand-side according to <ref>. In <ref>, we can replace $(\x, D)$ by $(A, B)$, as long as $[B, A] = 1$. To generalize the result for $[B, A] = p(A)$, we define $\pi$ by $\pi'(t) = 1/p(t)$, to have, by <Ref>, $[B, \pi(A)] = 1$, thus we can apply the less general result proved just before. We will do so by replacing $(f, g, A)$ by $(f \circ \pi^{-1}, g \circ \pi^{-1}, \pi(A))$. By choosing wisely the constant term of $\pi$, we can make $\pi^{-1}$ a well-defined power series. If we set $\alpha_0 = \pi^{-1} \circ \alpha$, <ref> become

\[
\gamma(t, x) = \int_0^t g(\alpha_0(\tau,x)) \odif\tau, \qquad \phi(t, x) = \int_0^t f(\alpha_0(\tau,x)) \odif\tau.
\]

$\alpha_0(0) = \pi^{-1}(\alpha(0)) = \pi^{-1}(\pi(A)) =  A$, and by the chain rule and <ref>

\[
\odv{\alpha_0}{t} = (\pi^{-1})'(\alpha) \odv{\alpha}{t} = \inv{\pi'(\pi^{-1}(\alpha))} (f \circ \pi^{-1})(\alpha) = p(\alpha_0) f(\alpha_0),
\]

which concludes the proof.

We now provide the most important instances of the theorem. Identification of the coefficients would respectively yield the BT and the expansion of $(AB)^n$.

If $[B, A] = p(A)$ then

\[
e^{(A+B)t} = \exp\bra{\int_0^t \alpha(\tau) \odif\tau} e^{B t},
\]

where $\alpha$ is the solution to the Cauchy problem

\[
\odv{\alpha}{t} = p(\alpha), \quad \alpha(0) = A.
\]

If $[B, A] = p(A)$ then

\[
e^{ABt} = \exp\bra{\int_0^t \alpha(\tau) \odif\tau}^B,
\]

where $\alpha$ is the solution to the Cauchy problem

\[
\odv{\alpha}{t} = \alpha p(\alpha), \quad \alpha(0) = A.
\]

The antinormal form of the theorem was also given by Viskov [22].

Let $p, f, g$ be formal power series and $A, B \in \A$ such that $[B, A] = p(B)$. Then

\[
\exp((Af(B) + g(B))t) = (e^A)^{\phi} e^\gamma,
\]

\[
\phi = \int_0^t f(\beta(\tau)) \odif\tau, \qquad \gamma = \int_0^t g(\beta(\tau)) \odif\tau,
\]

where $\beta$ is the solution to the Cauchy problem

\[
\odv{\beta}{t} = p(\beta) f(\beta), \quad \beta(0) = B.
\]

Viskov [22] claimed that the proof was similar, though this assertion doesn't immediately appear evident, and even seems rather untrue. Nonetheless, with the tools we built, this theorem now emerges as a mere consequence of the preceding ones, sparing us the need to prove the entire theorem anew.

We begin by applying the properties of the operator $\L$ defined in <Ref> to the equality stated in <Ref>. On the left-hand side, we have

\[
\L(\exp((f(A)B + g(A))t)) = \exp(\L(f(A)B + g(A))t) = \exp(C f(D) + g(D))t),
\]

(still with $[D, C] = p(D)$) while on the right-hand side

\[
\L(e^\gamma (e^\phi)^B) = \L((e^\phi)^B) \L(e^\gamma) = (e^D)^{\L(\phi)} e^{\L(\gamma)}.
\]

Hence the equality

\[
\exp(C f(D) + g(D))t) = (e^D)^{\L(\phi)} e^{\L(\gamma)},
\]

which is essentially a renaming of variables from the original statement.

And once again, the important instances are given below.

If $[B, A] = p(B)$ then

\[
e^{(A+B)t} = e^{A t} \exp\bra{\int_0^t \beta(\tau) \odif\tau},
\]

where $\beta$ is the solution to the Cauchy problem

\[
\odv{\beta}{t} = p(\beta), \quad \beta(0) = B.
\]

If $[B, A] = p(B)$ then

\[
e^{ABt} = (e^A)^{{\textstyle\int}_{\mkern-6mu0}^t \beta(\tau) \odif\tau},
\]

where $\beta$ is the solution to the Cauchy problem

\[
\odv{\beta}{t} = \beta p(\beta), \quad \beta(0) = B.
\]

§ COMMUTATOR OF THE FORM OF A MONOMIAL

In this section, we explore the normal ordered form of $(AB)^n$ and $(A+B)^n$ when the commutator is a monomial in $A$. While the former has been previously examined in the case where $[B, A] = h A^s$, we will now demonstrate a closely related formula for the binomial theorem when $[B, A] = h A^{s+1}$, utilizing Viskov's theorem.

§.§ Normal ordered form of (AB)^n

Mansour, Schork and Shattuck proved in [12] that if $A$ and $B$ are generators of the generalized Weyl algebra [10], that is, $[B, A] = h A^s$, then we could express

\begin{equation}\label{e:MSS}
(AB)^n = \sum_{k=0}^n \St_s(n,k) h^{n-k} A^{s(n-k)+k} B^k,
\end{equation}

where $\St_s$ represents the generalized Stirling number, defined as follows:

\begin{equation}
\St_0(n,k) = \Stir{n}{k}, \quad \St_1(n,k) = \coeff{n}{k},
\end{equation}

and for $s \in \K \setminus \{0, 1\}$

\begin{align}
\St_s(n,k)
&= \frac{n! s^n}{k! (1-s)^k} \sum_{j=0}^k (-1)^{k-j} \binom{k}{j} \binom{n + j(1/s -1) - 1}{n} \\
&= \inv{k!} \sum_{j=0}^k (-1)^{k-j} \binom{k}{j}\sum_{i=0}^n \coeff{n}{i} \frac{s^{n-i}j^i}{(1-s)^{k-i}}. \label{e:miss}
\end{align}

It appears that they overlooked the fact that by exchanging the sums in <ref> and utilizing <ref>, we could derive the improved closed form below, without any degenerate cases

\begin{equation}
\St_s(n,k) = \sum_{j=k}^n \coeff{n}{j} \Stir{j}{k} s^{n-j} (1-s)^{j-k}.
\end{equation}

Their proof employed the generating function

\begin{equation}\label{e:stirgengf}
\sum_{n=k}^\infty \St_s(n,k) \frac{t^n}{n!} = \frac{\mathfrak F_s(t)^k}{k!},
\end{equation}

\begin{equation}
\mathfrak F_s(t) \defeq \begin{cases}
e^t-1 &\com{if} s=0 \\
-\log(1-t) &\com{if} s=1 \\[3pt]
\dfrac{1-(1-st)^{1-1/s}}{s-1} &\com{else.}
\end{cases}
\end{equation}

We slightly adjusted the notation for $\St_s(n,k)$ to extract the dependency on $h$, and we corrected the lower index of the sum in <ref>. We set it to $0$ instead of $1$ because $\St_s(n,0) = \delta_n$. It is generally good practice to let the lower index be $0$ to also include the trivial case $n=0$.

The obvious examples are, of course, the cases $s = 0, 1$ but another interesting one is the case $s = 1/2$. Indeed, the formula $\St_{1/2}$ simplifies into Lah numbers $\Lah{n}{k} \defeq \binom{n-1}{k-1} \frac{n!}{k!}$ <cit.>, indeed

\begin{equation}
\St_{1/2}(n,k) =  2^{k-n} \sum_{j=k}^n \coeff{n}{j} \Stir{j}{k} = 2^{k-n} \Lah{n}{k},
\end{equation}

where we used an identity that can be easily proved by composing the Stirling generating functions (<ref>). We can rewrite <ref> with $[B, A] = 2 h \sqrt A$ into

\begin{equation}
(AB)^n = \sum_{k=0}^n \Lah{n}{k} h^{n-k} A^{\tfrac{n+k}{2}} B^k.
\end{equation}

The exponential version of the theorem proved by Mansour, Schork and Shattuck is the following.

If $[A, B] = h A^s$ then

\[
e^{ABt} = \exp\bra{\frac{\mathfrak F_s(hA^st)}{hA^{s-1}}}^B.
\]

By <ref>,

\begin{align*}
&= \sum_{n=0}^\infty \frac{t^n}{n!} (AB)^n = \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{k=0}^n \St_s(n,k) h^{n-k} A^{s(n-k)+k} B^k \\
&= \sum_{k=0}^\infty \bra{\sum_{n=k}^\infty \frac{t^n}{n!} \St_s(n,k) h^n A^{sn}} h^{-k} A^{(1-s)k} B^k,
\end{align*}

now we use <ref> and <Ref>

\[
e^{ABt} =  \sum_{k=0}^\infty \frac{\mathfrak F_s(hA^st)^k}{k!} h^{-k} A^{(1-s)k} B^k = \exp\bra{\frac{\mathfrak F_s(hA^st)}{hA^{s-1}}}^B.
\]

§.§ Binomial theorem

The EI was found in <cit.> with <Ref>, but the connection with generalized Stirling numbers was missed.

For $h, s \in \K$, these statements are equivalent

* []$[B, A] = h A^{s+1}$

* for $n \in \N$
\begin{align*}
&= \sum_{k=0}^n \binom{n}{k} \bra{\sum_{j=0}^k \St_s(k,j) h^{k-j} A^{s(k-j)+j}}B^{n-k} \\
&= \sum_{k=0}^n h^{n-k} \sum_{j=0}^k \binom{n}{j} \St_s(n-j, k-j) A^{s(n-k)+k-j} B^j
\end{align*}

* for $t \in \K$
\[
e^{(A+B)t} =  \exp\bra{\frac{\mathfrak F_s(hA^st)}{hA^{s-1}}} e^{Bt}.
\]

Suppose that $[B, A] = h A^s$. Then, by <Ref>,

\[
e^{ABt} = \exp\bra{\int_0^t \alpha(\tau) \odif\tau}^B,
\]

\[
\odv{\alpha}{t} = h \alpha^{s+1}, \quad \alpha(0) = 1. \tag{$*$}
\]

But by <Ref>, we can identify

\begin{equation}\label{e:identify}
\int_0^t \alpha(\tau) \odif\tau = \frac{\mathfrak F_s(hA^st)}{hA^{s-1}}.
\end{equation}

Now suppose i, that is $[B, A] = h A^{s+1}$, then by <Ref>,

\[
e^{(A+B)t} = \exp\bra{\int_0^t \alpha(\tau) \odif\tau} e^{Bt},
\]

with $\alpha$ still defined by $(*)$. Thus, we can make use of <ref> to obtain iii. ii follows from <ref>.

We could take advantage of the same trick used in the proof to derive, for example, one of the following

* the formula for $(AB)^n$, when $A$ and $B$ are generators of the generalized Ore algebra, i.e., when $[B, A] = \alpha + \epsilon A$ (found in [11])

* The BT when $[B, A] = A(\alpha + \epsilon A)$ (which is given by <Ref>)

using the other.

The formulas for $\frac{\mathfrak F_s(hA^st)}{hA^{s-1}}$ in the cases $s=-1, 0, 1$ are respectively the divisonless $At + h \frac{t^2}{2}$, $A \frac{e^{ht}-1}{h}$, and $-\log(1-hAt)$, while the other cases still involve division in their closed forms. Consequently, it's intriguing to observe that if $s+1 \notin \{0, 1, 2\}$, then the exponential identity cannot be expressed solely in terms of elementary functions without division (although they can in a power series form). This observation sheds light on the discussion in <Ref>, where we discussed how if the degree of the commutator isn't $0$, $1$, or $2$, then finding the exponential identity becomes more challenging. Thus, one of the reasons for this difficulty might be connected to the appearance of division by a power of $A$.

If $[B, A] = \tau A^3$,

\[
e^{(A+B)t} = e^{\tfrac{1-\sqrt{1-2 \tau A^2 t}}{\tau A}} e^{B t}.
\]

This case is associated with Bessel polynomials [14]. Indeed, according to Krall and Frink in [7], Bessel polynomials $\{y_n\}_{n\in\N}$, defined by

\[
y_n(x) \defeq \sum_{k=0}^n \frac{(n+k)!}{(n-k)!k!}\bra{\frac{x}{2}}^k = \sum_{k=0}^n \binom{n+k}{2k}(2k-1)!! \, x^k,
\]

satisfy this generating function formula

\[
\sum_{n=0}^\infty \frac{t^n}{n!} y_{n-1}(x)  = e^{\tfrac{1-\sqrt{1-2xt}}{x}}.
\]

Hence the formula for the EI (which requires no division)

\[
e^{(A+B)t} = \sum_{n=0}^\infty \frac{(At)^n}{n!} y_{n-1}(\tau A) e^{Bt}.
\]

Furthermore, the binomial theorem can be written as

\[
(A+B)^n = \sum_{k=0}^n \binom{n}{k} \bra{\sum_{j=0}^k \binom{k-1+j}{2j} (2j-1)!! \, \tau^j A^{k+j}} B^{n-k},
\]

which is at a change of index away from <Ref>, indicating that we have found the following closed form

\begin{equation}
\St_2(n,n-k) = \binom{n-1+k}{2k}(2k-1)!!.
\end{equation}

§ CONCLUSION

In our research, we have developed exponentiation and general tools for exploring noncommutative algebra. This approach enabled us to derive exponential identities and certain binomial theorems in cases where the commutator $[B, A] = f(A)$ takes the form of a polynomial of degree two or less. We also delved into the potential absence of a normal ordered form when the commutator involves two variables, providing illustrative examples to support our discussion.

We've shed light on an earlier theorem by Viskov [22], which serves as a powerful tool for solving numerous problems in normal ordering. The antinormal form was proved via the new operator $\L$. Viskov's theorem facilitated a connection between the exponential identity in cases where $f$ is a monomial of arbitrary degree and the normal ordered form of $(AB)^n$ when $f$ is one degree less, which was already documented [12, 13, 10]. This connection enabled the discovery of analogous formulas for both the exponential identity and the binomial theorem.

Further generalizing these findings by increasing the degree of the arbitrary polynomial $f$ appears to be a challenging task, as discussed. Additionally, exploring the case of a bivariate commutator remains an intriguing yet somewhat enigmatic area of study. Extending the work outlined in Section <Ref> and in previous literature [17, 8, 19, 6] could provide valuable insights into noncommutative algebra.

Moreover, now that we better understand regular normal ordering, investigating $q$-analogues seems to be another avenue for exploration. Investigating binomial theorems under conditions $BA - qAB = f(A)$, as studied in [1, 9, 10, 8], could offer further depth to our understanding of $q$-calculus and noncommutative algebra.

§.§.§ Acknowledgement

I would like to thank M. Schork for his precious advice and for sharing one of the references.

[1]
H.B. Benaoum.
($q,h$)-analogue of Newton's binomial formula.
J. Phys. A, 31(46):L751–L754, nov 1998.

[2]
M.V. Berry.
The Diffraction of Light by Ultrasound.
University of St Andrews thesis. Academic Press, 1966.

[3]
A. Di Bucchianico and D.E. Loeb.
Operator expansion in the derivative and multiplication by $x$.
Integral Transforms Spec. Funct, 4(1-2):49–68, 1996.

[4]
E. Clark and R. Ehrenborg.
The excedance algebra.
Discrete Mathematics, 313(13):1429–1435, 2013.

[5]
L. Comtet.
Advanced Combinatorics: The Art of Finite and Infinite
Springer Netherlands, 1974.

[6]
L. Dupays and J.-C. Pain.
Closed forms of the Zassenhaus formula.
J. Phys. A, 56(25):255202, may 2023.

[7]
H. Krall and O. Frink.
A new class of orthogonal polynomials: The Bessel polynomials.
Trans. Amer. Math. Soc, 65:100–115, 1949.

[8]
V. Levandovskyy, C. Koutschan, and O. Motsak.
On two-generated non-commutative algebras subject to the affine
In Computer Algebra in Scientific Computing, pages 309–320.
Springer Berlin Heidelberg, 2011.

[9]
T. Mansour and M. Schork.
The commutation relation $xy=qyx+hf(y)$ and Newton’s binomial
Ramanujan J., 25:405–445, apr 2011.

[10]
T. Mansour and M. Schork.
Commutation relations, normal ordering, and Stirling numbers.
CRC Press, jan 2015.

[11]
T. Mansour and M. Schork.
On Ore-Stirling numbers defined by normal ordering in the Ore
Filomat, 37(18):6115–6131, 2023.

[12]
T. Mansour, M. Schork, and M. Shattuck.
On a new family of generalized Stirling and Bell numbers.
Electron. J. Combin., 18:P77, mar 2011.

[13]
T. Mansour, M. Schork, and M. Shattuck.
The generalized Stirling and Bell numbers revisited.
J. Integer Seq., Article 12.8.3:pp. 47, oct 2012.

[14]
T. Mansour, M. Schork, and M. Shattuck.
On the Stirling numbers associated with the meromorphic Weyl
Appl. Math. Lett., pages 1767–, feb 2012.

[15]
M. Müger.
Notes on the theorem of Baker-Campbell-Hausdorff-Dynkin,

[16]
F. Olver, D. Lozier, R. Boisvert, and C. Clark.
The NIST Handbook of Mathematical Functions.
Cambridge University Press, New York, NY, may 2010.

[17]
H. Rosengren.
A non-commutative binomial formula.
J. Geom. Phys., 32:349–363, feb 2000.

[18]
R. A. Sack.
Taylor's theorem for shift operators.
Phil. Mag., 3(29):497–503, 1958.

[19]
A. Van-Brunt and M. Visser.
Special-case closed form of the Baker–Campbell–Hausdorff
J. Phys. A, 48(22):225207, may 2015.

[20]
O.V. Viskov.
On the ordered form of a non-commutative binomial.
Russian Math. Surveys, 46(2):258, apr 1991.

[21]
O.V. Viskov.
On the R.A. Sack theorem for translation operators.
Dokl. Akad. Nauk, 340:463–466, jan 1995.

[22]
O.V. Viskov.
Expansion in powers of a noncommutative binomial.
Tr. Mat. Inst. Steklova, 216:70–75, 1997.

[23]
O.V. Viskov.
An approach to ordering.
Dokl. Akad. Nauk, 57:69–71, jan 1998.

[24]
K. Yamazaki.
Operator Calculus in Quantized Field Theory.
Prog. Theor. Phys., 7(5-6):449–468, may 1952.


# Separability of Klein-Gordon Equation on Near Horizon Extremal
Myers-Perry Black Hole

Hovhannes Demirchian<EMAIL_ADDRESS>Yerevan Physics Institute, 2 Alikhanian
Brothers St., 0036, Yerevan, Armenia. Saeedeh Sadeghian<EMAIL_ADDRESS>Department of Theoretical Physics, Faculty of Basic Sciences, University of
Mazandaran, P.O. Box 47416-95447, Babolsar, Iran  ICRANet-Mazandaran,
University of Mazandaran, P. O. Box 47415-416, Babolsar, Iran.

###### Abstract

We investigate the separability of Klein-Gordon equation on near horizon of
d-dimensional rotating Myers-Perry black hole in two limits : 1) generic
extremal case and 2) extremal vanishing horizon case. In the first case ,
there is a relation between the mass and rotation parameters so that black
hole temperature vanishes. In the latter case, one of the rotation parameters
is restricted to zero on top of the extremality condition. We show that the
Klein-Gordon equation is separable in both cases. Also, we solved the radial
part of that equation and discuss its behaviour in small and large r regions.

††preprint: APS/123-QED

##  I Introduction

The four dimensional Kerr black hole has been widely studied from various
aspects. From the geometric point of view, Carter showed that it has
integrable geodesics[1]. The extension to higher dimensions is known as Myers-
Perry black hole. In d-dimensions, it is described by one mass parameter and
$N(=[\frac{d}{2}])$ number of rotation parameters. It has integrable geodesic
equations just like its 4-dimensional equivalent. Other extensions such as
including the cosmological constant and NUT charge does not change this
behavior[2, 3, 4]. The integrability of the Klein-Gordon and Maxwell field
equations have been also studied on the d-dimensional Kerr-(A)dS-NUT space-
time[2, 3, 5, 6, 7, 8].

One can construct another solution to Einstein equations in the Near Horizon
extremal limit [9, 10, 11, 12] of MP (NHEMP) black hole [13] (see [14, 15, 16,
17, 18, 19] for recent studies). The extremal limit of the parameters
describes a black hole with the biggest allowed angular momentum for a given
BH mass. When one of the rotation parameters of the BH vanishes, we arrive to
yet another solution of Einstein equations referred to as extremal vanishing
horizon (EVH) geometry[20, 21]. Although the generic extremal limit of Myers-
Perry black hole exists in both even and odd dimensions, the special EVH limit
exists only in odd-dimensions.

A set of the Killing vectors of NHEMP obey the structural relation of
$SL$(2,R) algebra corresponding to $AdS_{2}$ subspace. It has been
demonstrated (e.g. [22, 23, 24, 25]) that the Casimir element of this $SO$(2,
1) algebra gives rise to a reduced Hamiltonian system called spherical or
angular mechanics, which contains all the necessary information about the near
horizon geometry. In other words, a massive particle moving in the near
horizon geometry of an extremal rotating black hole possesses a dynamical
conformal symmetry, i.e. defines “conformal mechanics” [17, 16, 14, 18, 25,
26, 24, 27, 28, 29, 30, 31], whose Casimir element can be viewed as a reduced
Hamiltonian, which contains all the necessary information about the whole
system.

An eye-catching difference between NHEMP and its vanishing horizon limit is
that the latter has a larger isometry group[32, 33, 34]: it includes two
copies of SL(2,R) corresponding to the $AdS_{3}$ subspace instead of one copy
of SL(2,R) for the $AdS_{2}$ factor of the metric. This symmetry enhancement
does not add to number of independent constants of motion though [15].

The near horizon geometry of Myers-Perry black holes contains integrable and
superintegrable systems like Rosochatius and Pöschl-Teller which are
interestingly related to Klein-Gordon equation through a geometrization
procedure [35]. The angular part of near horizon limit of fully isotropic
Myers-Perry black hole is a superintegrable mechanics called Rosochatius
system. This is a direct generalization of the Higgs oscillator. Separation of
variables in Rosochatius system results into a recursive family of one-
dimensional Pöschl-Teller system. The quantum equivalents of Higgs oscillator,
Rosochatius and Pöschl-Teller systems can be associated with a Klein-Gordon
equation on a static spacetime.

In this work, we study the separability of Klein-Gordon equation on near
horizon geometries of odd-dimensional Myers-Perry black hole. First, we take
the near horizon geometry in the generic extremal case and use the elliptical
coordinates in which the geodesic equation is separable. The
SL(2,R)$\times$U(1)N isometry group of the background metric helps us to
simply separate the part of Klein-Gorodon equation related to $AdS_{2}$
subspace and azimuthal angles. We observe that the Klein-Gordon equation is
separable in the elliptical coordinates. We also find the solution to the
radial part of the Klein-Gordon equation and discuss some general properties
of the solution. Then, we study the near horizon EVH space-time and show the
separability of the Klein-Gordon equation on that metric.

##  II Klein-Gordon equation on near horizon extremal geometry

The near horizon extremal metric of odd dimensional $(d=2N+1)$ Myers-Perry
black hole (NHEMP) in Gaussian null coordinates was given in [13] and can be
written in the Boyer-Lindquist coordinates as

$\displaystyle ds^{2}=$
$\displaystyle\frac{F_{H}}{b(r_{H})}\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\right)$
(1)
$\displaystyle+\sum_{i=1}^{N}(r_{H}^{2}+a_{i}^{2})d\mu_{i}^{2}+\sum_{i,j=1}^{N}\gamma_{ij}D\varphi^{i}D\varphi^{j}\,,$

where,

$D\varphi^{i}\equiv d\varphi^{i}+\frac{B^{i}}{b}rd\tau,\qquad
B^{i}=\frac{2}{r_{H}^{2}}\frac{\sqrt{m_{i}-1}}{m_{i}^{2}}\,.$ (2)

The metric functions are

$\begin{gathered}F_{H}=\sum_{i=1}^{N}\frac{\mu_{i}^{2}}{m_{i}},\qquad
b(r_{H})=\frac{4}{r_{H}^{2}}\sum_{i<j}^{N}\frac{1}{m_{i}}\frac{1}{m_{j}}\,,\\\
\gamma_{ij}=(r_{H}^{2}+a_{i}^{2})\mu_{i}^{2}\delta_{ij}+\frac{1}{F_{H}}a_{i}\mu_{i}^{2}a_{j}\mu_{j}^{2}\,,\end{gathered}$
(3)

Here, $m_{i}$’s are some constant parameters related to the horizon radius
($r_{H}$) and $N$ number of rotation parameters ($a_{i}$’s) corresponding to
azimuthal coordinates $\varphi^{i}$ defined by,

$m_{i}=\frac{r_{H}^{2}+a_{i}^{2}}{r_{H}^{2}}\geq 1\,,$ (4)

and $\mu_{i}$ are the latitudinal coordinates which satisfy the following
relation

$\sum_{i=1}^{N}\mu_{i}^{2}=1\,.$ (5)

The location of the horizon $r_{H}$ is determined by the largest positive
solution of

$\Pi(r_{H})=m\,r_{H}^{2},$ (6)

where $\Pi(r)\equiv\prod_{i=1}^{N}(r^{2}+a_{i}^{2})$ and $m$ is a constant
related to the mass of the Myers-Perry black hole.

The extremal limit is given by

$\Pi^{\prime}(r_{H})=2m\,r_{H}\,.$ (7)

By combining these two relations one finds that $m_{i}$’s are restricted by

$\sum_{i=1}^{N}\frac{1}{m_{i}}=1\,.$ (8)

By substituting $\mu_{i}=x_{i}/\sqrt{m_{i}}$ for $i=1\ ...\ N$, the new form
of NHEMP metric in an arbitrary odd dimensions becomes

$\displaystyle\frac{ds^{2}}{r^{2}_{H}}=$ $\displaystyle
A(x)\left(-r^{2}d\tau^{2}+\frac{dr^{2}}{r^{2}}\right)$ (9)
$\displaystyle+\sum_{i=1}^{N}dx_{i}dx_{i}+\sum_{i,j=1}^{N}\tilde{\gamma}_{ij}x_{i}x_{j}D\varphi^{i}D\varphi^{j}\,,$

where

$\displaystyle D\varphi^{i}\equiv d\varphi^{i}+k^{i}rd\tau,\qquad
k^{i}=\frac{B^{i}}{b}\,,$ (10) $\displaystyle
A(x)=\frac{\sum_{i=1}^{N}x^{2}_{i}/m^{2}_{i}}{r_{H}^{2}\
b},\qquad\sum_{i=1}^{N}\frac{x^{2}_{i}}{m_{i}}=1\,,$
$\displaystyle\tilde{\gamma}_{ij}=\delta_{ij}+\frac{1}{\sum_{l}^{N}x_{l}^{2}/m^{2}_{l}}\frac{\sqrt{m_{i}-1}x_{i}}{m_{i}}\frac{\sqrt{m_{j}-1}x_{j}}{m_{j}}\,.$

The behavior of a massive scalar field $\Phi$ in the gravitational background
is governed by the Klein-Gordon equation:

$\square\Phi=\frac{1}{\sqrt{-g}}{\partial_{\alpha}(\sqrt{-g}g^{\alpha\beta}\partial_{\beta}\Phi)}=M^{2}\Phi\,,$
(11)

where $M$ is the mass of the scalar field and $g$ is the determinant of the
metric. Separation of variables of Klein-Gordon equation in the background of
odd dimensional NHEMP can be carried out in elliptic coordinates $\lambda_{a}$
which is related to $x_{i}$ with the following relation

$x^{2}_{i}=(m_{i}-\lambda_{i})\prod_{j=1,j\neq
i}^{N}\frac{m_{i}-\lambda_{j}}{m_{i}-m_{j}}\,,$ (12)

where $\lambda_{N}<m_{N}<\ldots<\lambda_{2}<m_{2}<\lambda_{1}<m_{1}$. To
resolve the last relation in (10) one should choose $\lambda_{N}=0$. In these
coordinates $\lambda_{a}$

$\sum_{i=1}^{N}dx_{i}^{2}=\sum_{a=1}^{N-1}h_{a}(\lambda)d\lambda_{a}^{2}\,,$
(13)

with

$h_{a}=-\frac{\lambda_{a}\prod_{b\neq
a}^{N-1}(\lambda_{b}-\lambda_{a})}{4\prod_{i=1}^{N}(m_{i}-\lambda_{a})}\,.$
(14)

the NHEMP metric (9) becomes

$\displaystyle\frac{ds^{2}}{r^{2}_{H}}=$ $\displaystyle
A(\lambda)\left(-r^{2}d\tau^{2}+\frac{dr^{2}}{r^{2}}\right)+\sum_{a=1}^{N-1}h_{a}(\lambda)d\lambda_{a}^{2}$
(15)
$\displaystyle+\sum_{i,j=1}^{N}\tilde{\gamma}_{ij}x_{i}(\lambda)x_{j}(\lambda)D\varphi^{i}D\varphi^{j},$

here

$A(\lambda)=\frac{1}{b}\frac{\prod\limits_{a=1}^{N-1}\lambda_{a}}{\prod\limits_{i=1}^{N}m_{i}}.$
(16)

To analysis the Klein-Gordon equation (11), we need to find the determinant
and the inverse metric of (15). The inverse of NHEMP metric is

$\displaystyle r^{2}_{H}\left(\frac{\partial}{\partial s}\right)^{2}=$
$\displaystyle-\frac{1}{A(\lambda)r^{2}}\left(\frac{\partial}{\partial\tau}-\sum_{i=1}^{N}r\
k^{i}\frac{\partial}{\partial\varphi^{i}}\right)^{2}$ (17)
$\displaystyle+\frac{r^{2}}{A(\lambda)}\left(\frac{\partial}{\partial
r}\right)^{2}+\sum_{a=1}^{N-1}h^{a}(\lambda)\left(\frac{\partial}{\partial\lambda_{a}}\right)^{2}$
$\displaystyle+\sum_{i,j=1}^{N}\tilde{\gamma}^{ij}\frac{1}{x_{i}(\lambda)}\frac{1}{x_{j}(\lambda)}\frac{\partial}{\partial\varphi^{i}}\frac{\partial}{\partial\varphi^{j}}\,,$

where $h^{a}(\lambda)$ and $\tilde{\gamma}^{ij}$ are the inverses of
$h_{a}(\lambda)$ and $\tilde{\gamma}_{ij}$ respectively

$h^{a}(\lambda)=h_{a}^{-1}(\lambda),\qquad\tilde{\gamma}^{ij}=\delta^{ij}-x_{i}\frac{\sqrt{m_{i}-1}}{m_{i}}x_{j}\frac{\sqrt{m_{j}-1}}{m_{j}}.$
(18)

The determinant of metric (15) has the following form

$-\det{g}=A(\lambda)r^{2}\cdot\frac{A(\lambda)}{r^{2}}\cdot\prod_{a=1}^{N-1}h_{a}\cdot
det\left(\tilde{\gamma}_{ij}x_{i}x_{j}\right).$ (19)

Taking into account the definition of $h_{a}$ (13) and the relation (12),
$\prod_{a}h_{a}$ simplifies to

$\displaystyle\prod_{a=1}^{N-1}h_{a}=c\cdot\frac{A(\lambda)\left(\prod\limits_{a<b}^{N-1}(\lambda_{b}-\lambda_{a})\right)^{2}}{\prod\limits_{i=1}^{N}x_{i}^{2}}$
, $\displaystyle
c=\frac{(-1)^{\frac{N(N-1)}{2}}b\left(\prod\limits_{i=1}^{N}m_{i}\right)^{2}}{4^{N-1}\prod\limits_{\begin{subarray}{c}i,j=1\\\
i\neq j\end{subarray}}^{N}(m_{i}-m_{j})}.$ (20)

Using matrix determinant lemma (70), the determinant of $\tilde{\gamma}_{ij}$
will take a simple form

$\det(\tilde{\gamma}_{ij}x_{i}x_{j})=\frac{1}{b\,A(\lambda)}\prod_{i=1}^{N}x_{i}^{2}$
(21)

Finally inserting (II) and (21) in (19), we find

$-\det{g}=c^{\prime}\left(\prod_{a=1}^{N-1}\lambda_{a}\right)^{2}\left(\prod\limits_{a<b}^{N-1}(\lambda_{b}-\lambda_{a})\right)^{2}\,,$
(22)

with

$\qquad
c^{\prime}=4^{1-N}\left(b\prod\limits_{i<j}^{N}(m_{i}-m_{j})\right)^{-2}$ (23)

Equipped with (17) and (22), we can rewrite the Klein-Gordon equation (11),
and noting two important relations

$\partial_{\lambda_{c}}\sqrt{-\det{g}}=\left(\frac{1}{\lambda_{c}}+\sum_{\begin{subarray}{c}b=1\\\
b\neq
c\end{subarray}}^{N-1}\frac{1}{\lambda_{c}-\lambda_{b}}\right)\sqrt{-\det{g}}\,,$
(24)
$\partial_{\lambda_{a}}h^{a}=\left(-\frac{1}{\lambda_{a}}-\sum_{i}^{N}\frac{1}{m_{i}-\lambda_{a}}-\sum_{\begin{subarray}{c}b=1\\\
b\neq a\end{subarray}}^{N-1}\frac{1}{\lambda_{a}-\lambda_{b}}\right)h^{a}\,,$
(25)

we rewrite the Klein-Gordon equation on NHEMP metric

$\displaystyle+\frac{1}{A(\lambda)}\left(-\frac{1}{r^{2}}\left[\frac{\partial}{\partial\tau}-\sum_{i=1}^{N}rk^{i}\frac{\partial}{\partial\varphi^{i}}\right]^{2}\varPhi+r^{2}\partial_{r}^{2}\varPhi+2r\partial_{r}\varPhi\right)$
$\displaystyle+\sum_{a=1}^{N-1}h^{a}\partial^{2}_{\lambda_{a}}\varPhi-\sum_{a=1}^{N-1}\sum_{i=1}^{N}\frac{h^{a}}{m_{i}-\lambda_{a}}\partial_{\lambda_{a}}\varPhi$
$\displaystyle+\sum_{i=1}^{N}\frac{1}{x_{i}^{2}}\partial^{2}_{\varphi_{i}}\varPhi-\sum_{i,j=1}^{N}\frac{\sqrt{m_{i}-1}}{m_{i}}\frac{\sqrt{m_{j}-1}}{m_{j}}\partial_{\varphi_{i}}\partial_{\varphi_{j}}\varPhi=M^{2}\varPhi\,.$

To separate the variables in this equation, we apply the following ansatz

$\varPhi=R_{r}(r)\cdot\prod_{a=1}^{N-1}R_{\lambda_{a}}(\lambda_{a})\cdot
e^{i\omega\tau}\cdot\prod_{b=1}^{N}e^{iL_{b}\varphi_{b}}\,,$ (27)

where $\omega$ and $L_{i}$ are arbitrary constants. In this form $\lambda_{a}$
and $r$ derivatives of the scalar $\varPhi$ will be

$\displaystyle\partial_{r}\varPhi=\frac{R^{{}^{\prime}}_{r}}{R_{r}}\varPhi,\qquad\partial^{2}_{r}\varPhi=\frac{R^{{}^{\prime\prime}}_{r}}{R_{r}}\varPhi\,,$
$\displaystyle\partial_{\lambda_{a}}\varPhi=\frac{R^{{}^{\prime}}_{\lambda_{a}}}{R_{\lambda_{a}}}\varPhi\,,\qquad\partial^{2}_{\lambda_{a}}\varPhi=\frac{R^{{}^{\prime\prime}}_{\lambda_{a}}}{R_{\lambda_{a}}}\varPhi\,,$
(28)

and the Klein-Gordon equation transforms into

$\displaystyle\left[\vphantom{\sum_{i,j}\frac{\sqrt{m_{i}-1}}{m_{i}}}\right.$
$\displaystyle\frac{1}{A(\lambda)}\left(r^{2}\frac{R^{{}^{\prime\prime}}_{r}}{R_{r}}+2r\frac{R^{{}^{\prime}}_{r}}{R_{r}}+\frac{1}{r^{2}}\,(\omega-r\sum_{i=1}^{N}k^{i}L_{i})^{2}\right)$
$\displaystyle+\sum_{a=1}^{N-1}h^{a}\frac{R^{{}^{\prime\prime}}_{\lambda_{a}}}{R_{\lambda_{a}}}-\sum_{a=1}^{N-1}\sum_{i=1}^{N}\frac{h^{a}}{m_{i}-\lambda_{a}}\frac{R^{{}^{\prime}}_{\lambda_{a}}}{R_{\lambda_{a}}}$
$\displaystyle\left.-\sum_{i=1}^{N}\frac{L^{2}_{i}}{x_{i}^{2}}+\sum_{i,j=1}^{N}\frac{\sqrt{m_{i}-1}}{m_{i}}\frac{\sqrt{m_{j}-1}}{m_{j}}L_{i}L_{j}\right]\varPhi=M^{2}\varPhi\,.$

One can see that the first term in the above equation only depends on $r$. The
separability requirement forces it to be a constant,

$r^{2}\frac{R^{{}^{\prime\prime}}_{r}}{R_{r}}+2r\frac{R^{{}^{\prime}}_{r}}{R_{r}}+\frac{1}{r^{2}}\,\left(\omega-r\sum_{i=1}^{N}k^{i}L_{i}\right)^{2}=\mathcal{C}_{2}\,.$
(31)

To write the radial equation (31) as a known differential equation, we change
the radial variable $r$ to $z$ by

$z=\frac{2\,i\,\omega}{r}\,,$ (32)

which brings the equation (31) into the familiar form of Whittaker’s equation

$\frac{d^{2}R_{r}}{dz^{2}}+\left(-\frac{1}{4}+\frac{K}{z}+\frac{(1/4-\mu^{2})}{z^{2}}\right)\,R_{r}=0\,,$
(33)

with

$K=i\sum_{j=1}^{N}k^{j}L_{j}\,,\quad\text{and}\quad\mu^{2}=\frac{1}{4}-\left(\sum_{i=1}^{N}k^{i}L_{i}\right)^{2}+\mathcal{C}_{2}\,.$
(34)

The general solutions to the equation (33) are Whittaker’s functions :
$\mathcal{M}_{K,\mu}(z)$ and $\mathcal{W}_{K,\mu}(z)$. These are related the
confluent hypergeometric functions.It is interesting to study the behaviour of
these functions in small and large $z$ region. (we keep $K,\mu$ parameter
fixed and generic.) For the small $z$ region, which is related to $r\to\infty$
region in the space-time, we have

$\displaystyle\mathcal{M}_{K,\mu}(z)\sim z^{\mu+1/2}\,,\quad\text{as $z\to
0$}$
$\displaystyle\mathcal{W}_{K,\mu}(z)\sim\alpha_{1}\,z^{\mu+1/2}+\alpha_{2}\,z^{-\mu+1/2}\,,\text{\footnotesize{
if $Re(\mu)<\frac{1}{2}$}}$
$\displaystyle\mathcal{W}_{K,\mu}(z)\sim\,z^{-\mu+1/2}\text{\footnotesize{ if
$Re(\mu)\geq\frac{1}{2}$}}$ (35)

here, $\alpha_{1,2}$ are some constants related to $\mu,K$. The asymptotic
behaviour ($z\to\infty$) of $\mathcal{M}_{K,\mu}(z)$ and
$\mathcal{W}_{K,\mu}(z)$ which is corresponding to $r\to 0$ region of space-
time is given as

$\displaystyle\mathcal{M}_{K,\mu}(z)\sim z^{-K}e^{z/2}$
$\displaystyle\mathcal{W}_{K,\mu}(z)\sim z^{K}e^{-z/2}\,,\quad\text{as
$z\to\infty$}$ (36)

Depending on the values of $\mu$ and $K$ in (34), the behaviour of the
solution is different : $\mathcal{M}_{K,\mu}$ is blowing up in the $r\to 0$
($z\to\infty$) limit, so we discard it. In the small z, i.e. ($r\to\infty$),
$\mathcal{W}_{K,\mu}$ is vanishing if $Re(\mu)<1/2$ which is guaranteed by
e.g. $\mathcal{C}_{2}<0$.

The rest of the variables can be separated after applying the relations (67),
(68) and (69) to the Klein-Gordon equation which will be transformed to

$\sum_{a=1}^{N-1}\frac{P_{\lambda_{a}}(\lambda_{a})-M^{2}(-\lambda_{a})^{N-2}}{\prod_{b=1;a\neq
b}^{N-1}(\lambda_{b}-\lambda_{a})}=0\,,$ (37)

where $P_{\lambda_{a}}(\lambda_{a})$ are defined by

$\displaystyle P_{\lambda_{a}}(\lambda_{a})\equiv$
$\displaystyle-\frac{4}{\lambda_{a}}\left(\frac{R^{{}^{\prime\prime}}_{\lambda_{a}}}{R_{\lambda_{a}}}-\frac{R^{{}^{\prime}}_{\lambda_{a}}}{R_{\lambda_{a}}}\sum_{i}^{N}\frac{1}{m_{i}-\lambda_{a}}\right)\prod_{j=1}^{N}(m_{j}-\lambda_{a})$
(38)
$\displaystyle+\frac{b}{\lambda_{a}}\mathcal{C}_{2}\prod\limits_{i=1}^{N}m_{i}+(-1)^{N-1}\sum_{i=1}^{N}\frac{g_{\varphi_{i}}}{m_{i}-\lambda_{a}}$
$\displaystyle+g_{0}(-\lambda_{a})^{N-2}\,.$

here $g_{\varphi_{i}}$ and $g_{0}$ are some constants, defined by

$g_{\varphi_{i}}\equiv\frac{k^{2}_{\varphi_{i}}}{m_{i}}\prod_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N}(m_{i}-m_{j})\,,\qquad
g_{0}\equiv\left(\sum_{i=1}^{N}\frac{\sqrt{m_{i}-1}}{m_{i}}L_{i}\right)^{2}\,.$
(39)

Equation (37) is only satisfied when

$P_{\lambda_{a}}(\lambda_{a})=\sum_{\alpha=1}^{N-1}k_{\alpha}\lambda_{a}^{\alpha-1},$
(40)

where $k_{\alpha}\ (\alpha=1,...,N-2)$ are arbitrary constants and
$k_{N-1}=(-1)^{N-2}M^{2}$.

##  III Klein-Gordon equation on near horizon EVH geometry

The metric of near horizon EVH Myers-Perry in the elliptical coordinates
parameterized with $(\tau,\rho,\psi,\lambda_{a},\varphi_{i})$, in $d=2N+1$
dimensions is of the form

$\displaystyle ds^{2}=$ $\displaystyle
F(\lambda)\left(-\rho^{2}d\tau^{2}+\frac{d\rho^{2}}{\rho^{2}}+\rho^{2}d\psi^{2}\right)$
$\displaystyle+\sum_{a=1}^{N-1}\hat{h}_{a}d\lambda_{a}^{2}+\sum_{a,b=1}^{N-1}\hat{\gamma}_{ab}\,\hat{x}_{a}(\lambda)\hat{x}_{b}(\lambda)d\varphi_{a}d\varphi_{b}\,,$

where the metric functions are

$\displaystyle
F(\lambda)=\prod_{a=1}^{N-1}\frac{\lambda_{a}}{m_{a}},\qquad\hat{h}_{a}=\frac{1}{4}\frac{{\prod_{{b\neq
a}}^{N-1}}\left(\lambda_{b}-\lambda_{a}\right)}{\prod_{c=1}^{N-1}\left(m_{c}-\lambda_{a}\right)}\,,$
$\displaystyle\hskip
28.45274pt\hat{\gamma}_{ab}=\delta_{ab}+\frac{1}{F(\lambda)}\frac{\hat{x}_{a}(\lambda)}{\sqrt{m_{a}}}\frac{\hat{x}_{b}(\lambda)}{\sqrt{m_{b}}}\,,$
(42)

with

$\displaystyle\hat{x}_{a}^{2}(\lambda)=\frac{\prod\limits_{b=1}^{N-1}(m_{a}-\lambda_{b})}{\prod\limits_{l\neq
a}^{N-1}(m_{a}-m_{l})}\,,\qquad\sum_{a=1}^{N-1}\frac{1}{m_{a}}=1.$ (43)

(We start this section in elliptical coordinates to avoid writing multiple
metrics. For more details, we refer the reader to [20, 14, 15].)

The inverse of metric (III) is

$\displaystyle\left(\frac{\partial}{\partial s}\right)^{2}=$
$\displaystyle\frac{1}{F(\lambda)}\left(-\frac{1}{\rho^{2}}(\frac{\partial}{\partial\tau})^{2}+\rho^{2}(\frac{\partial}{\partial\rho})^{2}+\frac{1}{\rho^{2}}(\frac{\partial}{\partial\psi})^{2}\right)$
(44)
$\displaystyle+\sum_{a=1}^{N-1}{\hat{h}}^{a}(\frac{\partial}{\partial\lambda_{a}})^{2}+\sum_{a,b=1}^{N-1}\frac{\hat{\gamma}^{ab}}{\hat{x}_{a}\hat{x}_{b}}\frac{\partial}{\partial\varphi^{a}}\frac{\partial}{\partial\varphi^{b}}\,,$

where

$\hat{\gamma}^{ab}=\delta_{ab}-\frac{\hat{x}_{a}(\lambda)}{\sqrt{m_{a}}}\frac{\hat{x}_{b}(\lambda)}{\sqrt{m_{b}}},\qquad\hat{h}^{a}=\left(\hat{h}_{a}\right)^{-1}\,.$
(45)

To study the Klein-Gordon equation on this metric,

$\square\Phi=\frac{1}{\sqrt{-g}}{\partial_{\alpha}(\sqrt{-g}g^{\alpha\beta}\partial_{\beta}\Phi)}=M^{2}\Phi\,,$
(46)

we need to compute the determinant of the metric,

$-\det{g}=\rho^{2}F_{0}(\lambda)^{3}\left(\prod_{a=1}^{N-1}\hat{h}_{a}\right)\det{(\hat{\gamma}_{ab}\hat{x}_{a}\hat{x}_{b})}\,,$
(47)

It is easy to show that

$\prod_{l=1}^{N-1}\hat{h}_{l}=\hat{c}\cdot\frac{\prod\limits_{\begin{subarray}{c}a,b=1\\\
a\neq
b\end{subarray}}^{N-1}\left(\lambda_{b}-\lambda_{a}\right)}{\prod\limits_{a=1}^{N-1}\hat{x}_{a}^{2}},\quad\hat{c}=4^{1-N}\prod_{\begin{subarray}{c}a,b=1\\\
a\neq b\end{subarray}}^{N-1}\left(m_{a}-m_{b}\right)^{-1}\,,$ (48)

Using matrix determinant lemma (70), we have

$\det{(\hat{\gamma}_{ab}\hat{x}_{a}\hat{x}_{b})}=\frac{1}{F(\lambda)}\prod_{a=1}^{N-1}\hat{x}_{a}^{2}\,.$
(49)

Finally, the determinant of metric (III) simplifies to

$-\det{g}=c^{\prime\prime}\,\rho^{2}\,\left(\prod_{l=1}^{N-1}\lambda_{l}\right)^{2}\left(\prod_{\begin{subarray}{c}a,b=1\\\
b<a\end{subarray}}^{N-1}(\lambda_{b}-\lambda_{a})\right)^{2}\,,$ (50)

with

$c^{\prime\prime}=\frac{1}{4^{N-1}}\left(\prod_{l=1}^{N-1}{m_{l}}\right)^{-2}\left(\prod_{\begin{subarray}{c}a,b=1\\\
b<a\end{subarray}}^{N-1}(m_{b}-m_{a})\right)^{-2}\,.$ (51)

The key relations for the separability of the Klein-Gordon equation are as
follows

$\displaystyle\partial_{\lambda_{a}}\sqrt{-\det{g}}=\left(\frac{1}{\lambda_{a}}-\sum_{\begin{subarray}{c}b=1\\\
b\neq
a\end{subarray}}^{N-1}\frac{1}{\lambda_{b}-\lambda_{a}}\right)\sqrt{-\det{g}}\,,$
(52)
$\displaystyle\partial_{\lambda_{a}}\hat{h}^{a}=\left(-\sum_{b=1}^{N-1}\frac{1}{m_{b}-\lambda_{a}}+\sum_{\begin{subarray}{c}b=1\\\
b\neq
a\end{subarray}}^{N-1}\frac{1}{\lambda_{b}-\lambda_{a}}\right)\hat{h}^{a}\,.$

The Klein-Gordon equation becomes

$\displaystyle\frac{1}{F(\lambda)}\left(-\frac{\partial_{\tau}^{2}\varPhi-\partial_{\psi}^{2}\varPhi}{\rho^{2}}+\rho^{2}\partial_{\rho}^{2}\varPhi+3\rho\partial_{\rho}\varPhi\right)$
$\displaystyle+\sum_{a=1}^{N-1}\frac{1}{\hat{x}_{a}^{2}}\partial^{2}_{\varphi_{a}}\varPhi-\sum_{a,b}^{N-1}\frac{1}{\sqrt{m_{a}}}\frac{1}{\sqrt{m_{b}}}\partial_{\varphi_{a}}\partial_{\varphi_{b}}\varPhi$
$\displaystyle+\sum_{a=1}^{N-1}\hat{h}^{a}\partial^{2}_{\lambda_{a}}\varPhi+\sum_{a=1}^{N-1}\frac{\hat{h}^{a}}{\lambda_{a}}\partial_{\lambda_{a}}\varPhi-\sum_{a,b=1}^{N-1}\frac{\hat{h}^{a}}{m_{b}-\lambda_{a}}\partial_{\lambda_{a}}\varPhi=M^{2}\varPhi\,.$

here $\hat{x}^{a}$ is defined in (43). To separate the variables, we use the
following ansatz for the scalar field

$\varPhi=R_{\rho}(\rho)\cdot\prod_{a=1}^{N-1}R_{a}(\lambda_{a})\cdot
e^{i(-k_{\tau}\tau+m_{\psi}\psi)}\cdot\prod_{b=1}^{N-1}e^{iL_{b}\varphi_{b}}\,,$
(54)

into the Klein-Gordon equation (III). We observe that the first parentheses
includes only $\rho$-dependent term. The separability requires it to be a
constant

$\left(\frac{k_{\tau}^{2}-m_{\psi}^{2}}{\rho^{2}}+\rho^{2}\partial_{\rho}^{2}+3\,\rho\,\partial_{\rho}\right)R_{\rho}(\rho)=-4\,\hat{\mathcal{C}}_{2}\,R_{\rho}(\rho)\,.$
(55)

Replacing

$\displaystyle
R_{\rho}(\rho)=\frac{u(\rho)}{\rho}\,,\quad\text{and}\quad\rho=\frac{\sqrt{k_{\tau}^{2}-m_{\psi}^{2}}}{z}\,,$
(56)

into the radial equation (55), we get the Bessel’s equation

$\displaystyle
z^{2}\frac{d^{2}}{dz^{2}}u+z\frac{d}{dz}u+\left(z^{2}-\nu^{2}\right)u=0$ (57)

with

$\displaystyle\nu^{2}=1-4\hat{\mathcal{C}}_{2}$ (58)

whose solutions are Bessel functions $J_{\nu}(z)$, $Y_{\nu}(z)$. The small and
large $r$ behaviour of the solution is dictated by the asymptotic behaviour of
Bessel functions. In the small $r$ region of space-time which is related to
$z\to\infty$, they give

$\displaystyle
J_{\nu}(z)\sim\frac{1}{\sqrt{z}}cos(z-\frac{\pi\nu}{2}-\frac{\pi}{4})\,,$
$\displaystyle
Y_{\nu}(z)\sim\frac{1}{\sqrt{z}}sin(z-\frac{\pi\nu}{2}-\frac{\pi}{4})\,,\text{as
$z\to\infty$}\,,$ (59)

which is vanishing. Also, in the large $r$ region of space-time, corresponding
to $z\to 0$, they behave as

$\displaystyle
J_{\nu}(z)\sim\frac{1}{\Gamma(\nu+1)}\left(\frac{z}{2}\right)^{\nu}\,,$
$\displaystyle
Y_{\nu}(z)\sim-\frac{\Gamma(\nu)}{\pi}\left(\frac{z}{2}\right)^{-\nu}\,.$ (60)

In the case of $\hat{\mathcal{C}}_{2}<0$ which leads to $\nu>1$, $Y_{\nu}$
blows up while $J_{\nu}$ falls off in the large $r$ region and is an
acceptable solution.

For the rest of the Klein-Gordon equation, we have

$\displaystyle\left(\vphantom{\sum_{i,j}\frac{\sqrt{m_{i}-1}}{m_{i}}}\right.$
$\displaystyle-\frac{4\hat{\mathcal{C}}_{2}}{F_{0}(\lambda)}-\sum_{a,b=1}^{N-1}\frac{L_{a}}{\sqrt{m_{a}}}\frac{L_{b}}{\sqrt{m_{b}}}$
$\displaystyle\left.+\sum_{a=1}^{N-1}\left[\hat{h}^{a}\frac{R^{{}^{\prime\prime}}_{a}}{R_{a}}+\frac{\hat{h}^{a}}{\lambda_{a}}\frac{R^{{}^{\prime}}_{a}}{R_{a}}-\sum_{i=1}^{N-1}\frac{\hat{h}^{a}}{m_{i}-\lambda_{a}}\frac{R^{{}^{\prime}}_{a}}{R_{a}}-\frac{k^{2}_{\varphi_{a}}}{x_{a}^{2}}\right]\right)\varPhi$
$\displaystyle\hskip 213.39566pt=M^{2}\varPhi\,.$

Using the relations (67), (68) and (69), we see that the Klein-Gordon equation
is separable

$\sum_{a=1}^{N-1}\frac{\hat{Q}_{\lambda_{a}}(\lambda_{a})-M^{2}(-\lambda_{a})^{N-2}}{\prod_{b=1;a\neq
b}^{N-1}(\lambda_{b}-\lambda_{a})}=0\,,$ (62)

where

$\displaystyle\hat{Q}_{\lambda_{a}}(\lambda_{a})=$
$\displaystyle\,4\left(\frac{R^{{}^{\prime\prime}}_{a}}{R_{a}}+\frac{1}{\lambda_{a}}\frac{R^{{}^{\prime}}_{a}}{R_{a}}\
-\sum_{b=1}^{N-1}\frac{R^{{}^{\prime}}_{a}/R_{a}}{m_{b}-\lambda_{a}}\right)\prod_{c=1}^{N-1}(m_{c}-\lambda_{a})$
(63)
$\displaystyle-\frac{4\hat{\mathcal{C}}_{2}}{\lambda_{a}}\prod\limits_{b=1}^{N-1}m_{b}+\sum_{b=1}^{N-1}\frac{\hat{q}_{\varphi_{b}}}{m_{b}-\lambda_{a}}+\hat{q}_{0}(-\lambda_{a})^{N-2}\,,$

and

$\hat{q}_{\varphi_{a}}\equiv(-1)^{N-1}\,k^{2}_{\varphi_{a}}\prod_{\begin{subarray}{c}b=1\\\
b\neq
a\end{subarray}}^{N-1}(m_{a}-m_{b})\,,\quad\hat{q}_{0}\equiv\left(\sum_{a=1}^{N-1}\frac{L_{a}}{\sqrt{m_{a}}}\right)^{2}\
.$ (64)

Equation (62) is only satisfied when

$\hat{Q}_{\lambda_{a}}(\lambda_{a})=\sum_{\alpha=1}^{N-1}k_{\alpha}\lambda_{a}^{\alpha-1},$
(65)

where $k_{\alpha}\ (\alpha=1,...,N-2)$ are arbitrary constants and
$k_{N-1}=(-1)^{N-2}M^{2}$.

##  IV Discussions

We studied the separability of Klein-Gordon equation on two near horizon
geometries in d-dimensions : generic extremal and extremal vanishing horizon
cases. Since the latter case exists only in odd dimensions, we take $d=2n+1$
in both cases. We do not expect that the separability of Klein-Gordon
equations changes for the first case if we take even-dimensions. This
expectation roots from the fact that the separability of geodesic equation has
been shown for both even and odd dimensions [15] and the Klein-gordon and
geodesic equation are related in the semi-classical limit : if we write the
solution to the Klein-Gordon equation $\Phi$ as
$\Phi=\mathcal{N}\,exp\left(\frac{iS}{\alpha}\right)$, we get the Hamilton-
Jacobi equation for $S$ in the so-called semi-classical limit, $\alpha\to 0$.
(see [36] for an explicit example.)

It is worth mentioning that since the extremal/EVH limit does not commute with
near horizon limit, we _could not_ get the Klein-Gordon equation on near
horizon EVH geometry by taking the EVH limit on NHEMP metric. Therefore, we
studied the Klein-Gordon equation on near horizon EVH geometry, independently.

One of the interesting problems to which we will come back in future is to
study the cases when some or all of the rotation parameters are equal.

Regarding the solution to the radial equations, we discussed the asymptotic
behaviour for the generic values of the parameters. There are some special
limits in the parameter space which can lead to different asymptotic behaviour
(like $\mu=0$ of $\mathcal{M}_{K,\mu}$ or $\mathcal{W}_{K,\mu}$). These
special limits are known for Whittaker and Bessel functions. We are interested
in the interpretation of their consequences in near horizon geometries.

Our next step is to study the other probes on these near horizon geometries
such as Dirac, Maxwell and p-form fields and show the separability of their
field equations.

## Acknowledgement

We would like to thank M.M. Sheikh-Jabbari and A. Nersessian for the fruitful
discussions during our previous collaboration, their contribution at the first
stages of this project and comments on the draft. H.D. acknowledges the
support of ICTP program Network scheme NT-04 for his visit to School of
Physics, IPM.

##  Appendix A Useful Identities

The following identity holds between $N-1$ independent variables $\lambda_{a}$
and any parameter $\kappa$

$\displaystyle\sum_{a=1}^{N-1}\frac{1}{\prod\limits_{\begin{subarray}{c}b=1\\\
{b\neq
a}\end{subarray}}^{N-1}(\lambda_{a}-\lambda_{b})}\frac{(\lambda_{a})^{\alpha}}{\kappa-\lambda_{a}}=\frac{(\kappa)^{\alpha}}{\prod\limits_{a=1}^{N-1}(\kappa-\lambda_{a})}-\delta_{\alpha,N-1}\,,$
(66)

where $\alpha=\\{0,...,N-1\\}$. For $\alpha=0$, this equation reduces to

$\frac{1}{\prod\limits_{a=1}^{N-1}(\lambda_{a}-\kappa)}=\sum_{a=1}^{N-1}\frac{1}{\prod\limits_{b=1;a\neq
b}^{N-1}(\lambda_{b}-\lambda_{a})}\frac{1}{\lambda_{a}-\kappa}$ (67)

and for an additional condition of $\kappa=0$ we get

$\frac{1}{\lambda_{1}\ldots\lambda_{N-1}}=\sum_{a=1}^{N-1}\frac{1}{\prod_{b=1;b\neq
a}^{N-1}(\lambda_{b}-\lambda_{a})}\frac{1}{\lambda_{a}}$ (68)

On the other hand, setting $\kappa=0$ in (66) results into

$\sum_{i=1}^{N}\frac{\lambda^{\beta}_{i}}{\prod_{j=1;i\neq
j}^{N}(\lambda_{i}-\lambda_{j})}=\delta_{\beta,N-1}\quad{\rm for}\quad
0\leq\beta\leq N-1.$ (69)

Another relation that we use in this paper is the so called matrix determinant
lemma which states

$\det(\mathrm{I}+\bm{xy}^{T})=1+\bm{y}^{T}\bm{x}$ (70)

where $\mathrm{I}$ is a unit matrix and $\bm{xy}^{T}$ is the outer product of
two vectors $\bm{x}$ and $\bm{y}$.

## References

  * [1] B. Carter, “Global structure of the Kerr family of gravitational fields,” Phys. Rev. 174 (1968) 1559–1571.
  * [2] V. P. Frolov, P. Krtous, and D. Kubiznak, “Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes,” JHEP 02 (2007) 005, hep-th/0611245.
  * [3] P. Krtouš, D. Kubizňák, D. N. Page, and M. Vasudevan, “Constants of geodesic motion in higher-dimensional black-hole spacetimes,” Phys. Rev. D 76 (oct, 2007) 084034.
  * [4] D. N. Page, D. Kubizňák, M. Vasudevan, and P. Krtouš, “Complete integrability of geodesic motion in general higher-dimensional rotating black-hole spacetimes,” Phys. Rev. Lett. 98 (Feb, 2007) 061102.
  * [5] M. Cariglia, P. Krtouš, and D. Kubizňák, “Dirac equation in Kerr-NUT-(A)dS spacetimes: Intrinsic characterization of separability in all dimensions,” Phys. Rev. D 84 (jul, 2011) 024008.
  * [6] I. Kolar and P. Krtous, “Spacetimes with a separable Klein-Gordon equation in higher dimensions,” Phys. Rev. D 93 (2016), no. 2, 024053, 1509.01667. [Addendum: Phys.Rev.D 93, 049903 (2016)].
  * [7] O. Lunin, “Maxwell’s Equations in the Myers-Perry Geometry,” 1708.06766.
  * [8] P. Krtouš, V. P. Frolov, and D. Kubizňák, “Separation of Maxwell equations in Kerr–NUT–(A)dS spacetimes,” Nucl. Phys. B 934 (sep, 2018) 7–38.
  * [9] J. Bardeen and G. T. Horowitz, “Extreme kerr throat geometry: A vacuum analog of ${\mathrm{ads}}_{2}{\times{}\mathrm{s}}^{2}$,” Phys. Rev. D 60 (Oct, 1999) 104030.
  * [10] H. K. Kunduri, J. Lucietti, and H. S. Reall, “Near-horizon symmetries of extremal black holes,” Class. Quantum Gravity 24 (aug, 2007) 4169–4189.
  * [11] H. K. Kunduri and J. Lucietti, “A classification of near-horizon geometries of extremal vacuum black holes,” J. Math. Phys. 50 (aug, 2009) 082502\.
  * [12] H. K. Kunduri and J. Lucietti, “Classification of Near-Horizon Geometries of Extremal Black Holes,” Living Rev. Relativ. 16 (2013).
  * [13] P. Figueras, H. K. Kunduri, J. Lucietti, and M. Rangamani, “Extremal vacuum black holes in higher dimensions,” Phys. Rev. D 78 (aug, 2008) 044042\.
  * [14] T. Hakobyan, A. Nersessian, and M. M. Sheikh-Jabbari, “Near horizon extremal Myers–Perry black holes and integrability of associated conformal mechanics,” Phys. Lett. B 772 (2017) 586–591, 1703.00713.
  * [15] H. Demirchian, A. Nersessian, S. Sadeghian, and M. Sheikh-Jabbari, “Integrability of geodesics in near-horizon extremal geometries: Case of Myers-Perry black holes in arbitrary dimensions,” Phys. Rev. D 97 (may, 2018) 104004.
  * [16] A. Galajinsky, A. Nersessian, and A. Saghatelian, “Action-angle variables for spherical mechanics related to near horizon extremal Myers–Perry black hole,” J. Phys. Conf. Ser. 474 (nov, 2013) 012019.
  * [17] A. Galajinsky, A. Nersessian, and A. Saghatelian, “Superintegrable models related to near horizon extremal Myers-Perry black hole in arbitrary dimension,” 1303.4901.
  * [18] H. Demirchian, “Note on constants of motion in conformal mechanics associated with near horizon extremal Myers–Perry black holes,” Mod. Phys. Lett. A 32 (2017), no. 27, 1750144, 1706.04861.
  * [19] I. Kolar and P. Krtous, “NUT-like and near-horizon limits of Kerr-NUT-(A)dS spacetimes,” Phys. Rev. D 95 (2017), no. 12, 124044, 1701.03950.
  * [20] J. de Boer, M. Johnstone, M. M. Sheikh-Jabbari, and J. Simón, “Emergent IR dual 2d CFTs in charged AdS 5 black holes,” Phys. Rev. D 85 (apr, 2012) 084039.
  * [21] M. M. Sheikh-Jabbari and H. Yavartanoo, “EVH Black Holes, AdS3 Throats and EVH/CFT Proposal,” 1107.5705.
  * [22] T. Hakobyan, S. Krivonos, O. Lechtenfeld, and A. Nersessian, “Hidden symmetries of integrable conformal mechanical systems,” 0908.3290.
  * [23] T. Hakobyan, O. Lechtenfeld, A. Nersessian, and A. Saghatelian, “Invariants of the spherical sector in conformal mechanics,” 1008.2912.
  * [24] A. Galajinsky and A. Nersessian, “Conformal mechanics inspired by extremal black holes in d=4,” 1108.3394.
  * [25] A. Galajinsky, “Particle dynamics on AdS_2 x S^2 background with two-form flux,” 0806.1629.
  * [26] A. Galajinsky, “Particle dynamics near extreme Kerr throat and supersymmetry,” 1009.2341.
  * [27] A. Galajinsky, “Near horizon black holes in diverse dimensions and integrable models,” 1209.5034.
  * [28] A. Galajinsky and K. Orekhov, “N=2 superparticle near horizon of extreme Kerr-Newman-AdS-dS black hole,” 1103.1047.
  * [29] S. Bellucci, A. Nersessian, and V. Yeghikyan, “Action-angle variables for the particle near extreme Kerr throat,” 1112.4713.
  * [30] A. Saghatelian, “Near-horizon dynamics of particle in extreme Reissner-Nordstr$\backslash$”om and Cl$\backslash$’ement-Gal’tsov black hole backgrounds: action-angle variables,” 1205.6270.
  * [31] A. Galajinsky and K. Orekhov, “On the near horizon rotating black hole geometries with NUT charges,” 1604.08056.
  * [32] S. Sadeghian, M. Sheikh-Jabbari, M. Vahidinia, and H. Yavartanoo, “Near horizon structure of extremal vanishing horizon black holes,” Nuclear Physics B 900 (2015) 222–243.
  * [33] S. Sadeghian, M. Sheikh-Jabbari, M. Vahidinia, and H. Yavartanoo, “Three theorems on near horizon extremal vanishing horizon geometries,” Physics Letters B 753 (2016) 488–492.
  * [34] S. Sadeghian and M. H. Vahidinia, “${\mathrm{ads}}_{3}$ to ${\mathrm{ds}}_{3}$ transition in the near horizon of asymptotically de sitter solutions,” Phys. Rev. D 96 (Aug, 2017) 044004.
  * [35] O. Evnin, H. Demirchian, and A. Nersessian, “Klein-Gordonization: mapping superintegrable quantum mechanics to resonant spacetimes,” 1711.03297.
  * [36] A. Sergyeyev and P. Krtous, “Complete Set of Commuting Symmetry Operators for Klein-Gordon Equation in Generalized Higher-Dimensional Kerr-NUT-(A)dS Spacetimes,” Phys. Rev. D 77 (2008) 044033, 0711.4623.