在數學 和物理學 中，馬格努斯展開 （英語：Magnus expansion ）為線性算子的一階齊次線性微分方程 的解提供了指數表示，得名於數學家威廉·馬格努斯 。特別地，這種方法提供了變係數n階線性常微分方程 組的基矩陣。指數是無窮級數，其項涉及多重積分和嵌套換元。
給定n × n A (t )
Y
′
(
t
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=
A
(
t
)
Y
(
t
)
,
Y
(
t
0
)
=
Y
0
{\displaystyle Y'(t)=A(t)Y(t),\quad Y(t_{0})=Y_{0}}
其中Y (t )
n = 1時，解為
Y
(
t
)
=
exp
(
∫
t
0
t
A
(
s
)
d
s
)
Y
0
.
{\displaystyle Y(t)=\exp \left(\int _{t_{0}}^{t}A(s)\,ds\right)Y_{0}.}
若A (t )t , t 1 、t 2 仍滿足A (t 1 ) A (t 2 ) = A (t 2 ) A (t 1 )n > 1情形。A t
馬格努斯提出的解矩陣初值問題的方法是，用某個n階方陣函數Ω(t , t 0 )
Y
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t
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=
exp
(
Ω
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t
0
)
)
Y
0
,
{\displaystyle Y(t)=\exp {\big (}\Omega (t,t_{0}){\big )}\,Y_{0},}
稍後可將其構造為級數 展開式：
Ω
(
t
)
=
∑
k
=
1
∞
Ω
k
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t
)
,
{\displaystyle \Omega (t)=\sum _{k=1}^{\infty }\Omega _{k}(t),}
為簡單起見習慣將Ω(t , t 0 ) Ω(t ) t 0 = 0.
馬格努斯意識到，由於d / dt e Ω ) e −Ω = A (t )Ω Ω 伴隨自同態 聯繫起來
Ω
′
=
ad
Ω
exp
(
ad
Ω
)
−
1
A
,
{\displaystyle \Omega '={\frac {\operatorname {ad} _{\Omega }}{\exp(\operatorname {ad} _{\Omega })-1}}A,}
並以「BCH展開 的連續類似物」遞歸求解Ω
上式構成了矩陣線性初值問題求解的馬格努斯展開式 或馬格努斯級數 。前4項：
Ω
1
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0
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A
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d
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1
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[
A
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,
Ω
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1
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1
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[
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+
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)
,
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(
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[
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+
[
A
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[
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[
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1
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]
)
,
{\displaystyle {\begin{aligned}\Omega _{1}(t)&=\int _{0}^{t}A(t_{1})\,dt_{1},\\\Omega _{2}(t)&={\frac {1}{2}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\,[A(t_{1}),A(t_{2})],\\\Omega _{3}(t)&={\frac {1}{6}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\,{\Bigl (}{\big [}A(t_{1}),[A(t_{2}),A(t_{3})]{\big ]}+{\big [}A(t_{3}),[A(t_{2}),A(t_{1})]{\big ]}{\Bigr )},\\\Omega _{4}(t)&={\frac {1}{12}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\int _{0}^{t_{3}}dt_{4}\,\left({\Big [}{\big [}[A_{1},A_{2}],A_{3}{\big ]},A_{4}{\Big ]}\right.\\&\qquad +{\Big [}A_{1},{\big [}[A_{2},A_{3}],A_{4}{\big ]}{\Big ]}+{\Big [}A_{1},{\big [}A_{2},[A_{3},A_{4}]{\big ]}{\Big ]}+\left.{\Big [}A_{2},{\big [}A_{3},[A_{4},A_{1}]{\big ]}{\Big ]}\right),\end{aligned}}}
其中[A , B ] ≡ A B − B A A 、B 的矩陣交換子 。
這些方程可解釋如下：Ω1 (t ) n 李群 ），則要對指數進行修正。馬格努斯級數的剩餘部分系統地提供了修正：Ω 李群 的李代數 中。
在應用中，很少能對馬格努斯級數精確求和，而要截斷才能得到近似解。馬格努斯方法的主要優勢在於，中截級數通常和精確解具有相同的重要性質，這異於傳統攝動理論 。例如，經典力學 中，時間演化 的辛幾何 特徵在每階近似都得到保留。同樣，量子力學 時間演化算子的幺正性 也得到保留（例如，與解決同一問題的戴森級數 相反）。
從數學角度看，收斂問題如下：給定矩陣A (t )Ω(t )
t ∈ [0,T )收斂 的充分條件是
∫
0
T
‖
A
(
s
)
‖
2
d
s
<
π
,
{\displaystyle \int _{0}^{T}\|A(s)\|_{2}\,ds<\pi ,}
其中
‖
⋅
‖
2
{\displaystyle \|\cdot \|_{2}}
矩陣範數 。這個結果是通用的，因為可構造特定矩陣，t > T
生成馬格努斯展開式中所有項的遞歸過程利用了下面的遞歸定義的矩陣 S n (k )
S
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Ω
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−
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j
−
1
)
]
,
2
≤
j
≤
n
−
1
,
{\displaystyle S_{n}^{(j)}=\sum _{m=1}^{n-j}\left[\Omega _{m},S_{n-m}^{(j-1)}\right],\quad 2\leq j\leq n-1,}
S
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1
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Ω
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S
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{\displaystyle S_{n}^{(1)}=\left[\Omega _{n-1},A\right],\quad S_{n}^{(n-1)}=\operatorname {ad} _{\Omega _{1}}^{n-1}(A),}
然後得到
Ω
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∫
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)
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{\displaystyle \Omega _{1}=\int _{0}^{t}A(\tau )\,d\tau ,}
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j
j
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2.
{\displaystyle \Omega _{n}=\sum _{j=1}^{n-1}{\frac {B_{j}}{j!}}\int _{0}^{t}S_{n}^{(j)}(\tau )\,d\tau ,\quad n\geq 2.}
此處adk Ω 是迭代交換子的簡寫（參見伴隨自同態 ）：
ad
Ω
0
A
=
A
,
ad
Ω
k
+
1
A
=
[
Ω
,
ad
Ω
k
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]
,
{\displaystyle \operatorname {ad} _{\Omega }^{0}A=A,\quad \operatorname {ad} _{\Omega }^{k+1}A=[\Omega ,\operatorname {ad} _{\Omega }^{k}A],}
其中B j 伯努利數 ，而B 1 = −1/2
最後，明確算得這一遞歸後，就可將Ωn t ) A 的n-1個嵌套換元的n重積分的線性組合：
Ω
n
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B
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⋯
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1
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t
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⋯
ad
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A
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n
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,
{\displaystyle \Omega _{n}(t)=\sum _{j=1}^{n-1}{\frac {B_{j}}{j!}}\sum _{k_{1}+\cdots +k_{j}=n-1 \atop k_{1}\geq 1,\ldots ,k_{j}\geq 1}\int _{0}^{t}\operatorname {ad} _{\Omega _{k_{1}}(\tau )}\operatorname {ad} _{\Omega _{k_{2}}(\tau )}\cdots \operatorname {ad} _{\Omega _{k_{j}}(\tau )}A(\tau )\,d\tau ,\quad n\geq 2,}
隨着n
要推廣到隨機常微分方程，令
(
W
t
)
t
∈
[
0
,
T
]
{\textstyle \left(W_{t}\right)_{t\in [0,T]}}
R
q
{\textstyle \mathbb {R} ^{q}}
布朗運動 ，
q
∈
N
>
0
{\textstyle q\in \mathbb {N} _{>0}}
概率空間
(
Ω
,
F
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P
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{\textstyle \left(\Omega ,{\mathcal {F}},\mathbb {P} \right)}
T
>
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{\textstyle T>0}
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{\displaystyle dX_{t}=B_{t}X_{t}dt+A_{t}^{(j)}X_{t}dW_{t}^{j},\quad X_{0}=I_{d},\qquad d\in \mathbb {N} _{>0},}
其中
B
⋅
,
A
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1
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,
…
,
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j
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{\textstyle B_{\cdot },A_{\cdot }^{(1)},\dots ,A_{\cdot }^{(j)}}
d
×
d
{\textstyle d\times d}
隨機過程 ，
I
d
{\textstyle I_{d}}
單位矩陣 。參考確定情形，並依隨機情形做修改[ 1]
Y
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+
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{\textstyle Y_{t}^{(1)}=Y_{t}^{(1,0)}+Y_{t}^{(0,1)}}
Y
t
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=
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0
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+
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+
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t
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{\textstyle Y_{t}^{(2)}=Y_{t}^{(2,0)}+Y_{t}^{(1,1)}+Y_{t}^{(0,2)}}
i j
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d
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{\displaystyle {\begin{aligned}Y_{t}^{(0,0)}&=0,\\Y_{t}^{(1,0)}&=\int _{0}^{t}A_{s}^{(j)}\,dW_{s}^{j},\\Y_{t}^{(0,1)}&=\int _{0}^{t}B_{s}\,ds,\\Y_{t}^{(2,0)}&=-{\frac {1}{2}}\int _{0}^{t}{\big (}A_{s}^{(j)}{\big )}^{2}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}A_{r}^{(i)}\,dW_{r}^{i}{\Big ]}dW_{s}^{j},\\Y_{t}^{(1,1)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}A_{r}^{(j)}\,dW_{r}{\Big ]}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}B_{r}\,dr{\Big ]}\,dW_{s}^{j},\\Y_{t}^{(0,2)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}B_{r}\,dr{\Big ]}\,ds.\end{aligned}}}
隨機情形下，收斂將受制於停止時間
τ
{\textstyle \tau }
[ 2]
X
=
(
X
t
)
t
∈
[
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,
T
]
{\textstyle X=(X_{t})_{t\in [0,T]}}
τ
≤
T
{\textstyle \tau \leq T}
X
t
{\textstyle X_{t}}
τ
{\textstyle \tau }
Y
t
{\textstyle Y_{t}}
X
t
=
e
Y
t
,
0
≤
t
<
τ
;
{\displaystyle X_{t}=e^{Y_{t}},\qquad 0\leq t<\tau ;}
以下表示有
P
{\textstyle \mathbb {P} }
Y
t
=
∑
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=
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∞
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t
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n
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≤
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<
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,
{\displaystyle Y_{t}=\sum _{n=0}^{\infty }Y_{t}^{(n)},\qquad 0\leq t<\tau ,}
其中
Y
(
n
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{\textstyle Y^{(n)}}
存在正常數C
‖
A
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‖
T
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‖
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{\textstyle \|A^{(1)}\|_{T},\dots ,\|A^{(q)}\|_{T},\|B\|_{T},T,d}
‖
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⋅
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t
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,
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)
{\textstyle \|A_{\cdot }\|_{T}=\|\|A_{t}\|_{F}\|_{L^{\infty }(\Omega \times [0,T])}}
P
(
τ
≤
t
)
≤
C
t
,
t
∈
[
0
,
T
]
.
{\displaystyle \mathbb {P} (\tau \leq t)\leq Ct,\qquad t\in [0,T].}
隨機馬格努斯展開的推廣形式：
Y
t
=
∑
n
=
0
∞
Y
t
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n
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with
Y
t
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n
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:=
∑
r
=
0
n
Y
t
(
r
,
n
−
r
)
,
{\displaystyle Y_{t}=\sum _{n=0}^{\infty }Y_{t}^{(n)}\quad {\text{with}}\quad Y_{t}^{(n)}:=\sum _{r=0}^{n}Y_{t}^{(r,n-r)},}
其中通用項
Y
(
r
,
n
−
r
)
{\textstyle Y^{(r,n-r)}}
Y
t
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r
,
n
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=
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0
t
μ
s
r
,
n
−
r
d
s
+
∫
0
t
σ
s
r
,
n
−
r
,
j
d
W
s
j
,
n
∈
N
0
,
r
=
0
,
…
,
n
,
{\displaystyle Y_{t}^{(r,n-r)}=\int _{0}^{t}\mu _{s}^{r,n-r}ds+\int _{0}^{t}\sigma _{s}^{r,n-r,j}dW_{s}^{j},\qquad n\in \mathbb {N} _{0},\ r=0,\dots ,n,}
σ
r
,
n
−
r
,
j
,
μ
r
,
n
−
r
{\textstyle \sigma ^{r,n-r,j},\mu ^{r,n-r}}
σ
s
r
,
n
−
r
,
j
:=
∑
i
=
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n
−
1
β
i
i
!
S
s
r
−
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,
n
−
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,
i
(
A
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,
μ
s
r
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n
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:=
∑
i
=
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n
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1
β
i
i
!
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s
r
,
n
−
r
−
1
,
i
(
B
)
−
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2
∑
j
=
1
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∑
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n
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2
β
i
i
!
∑
q
1
=
2
r
∑
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=
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n
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r
S
r
−
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,
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−
r
−
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,
i
(
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q
1
,
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2
,
j
)
,
{\displaystyle {\begin{aligned}\sigma _{s}^{r,n-r,j}&:=\sum _{i=0}^{n-1}{\frac {\beta _{i}}{i!}}S_{s}^{r-1,n-r,i}{\big (}A^{(j)}{\big )},\\\mu _{s}^{r,n-r}&:=\sum _{i=0}^{n-1}{\frac {\beta _{i}}{i!}}S_{s}^{r,n-r-1,i}(B)-{\frac {1}{2}}\sum _{j=1}^{q}\sum _{i=0}^{n-2}{\frac {\beta _{i}}{i!}}\sum _{q_{1}=2}^{r}\sum _{q_{2}=0}^{n-r}S^{r-q_{1},n-r-q_{2},i}{\big (}Q^{q_{1},q_{2},j}{\big )},\end{aligned}}}
其中
Q
s
q
1
,
q
2
,
j
:=
∑
i
1
=
2
q
1
∑
i
2
=
0
q
2
∑
h
1
=
1
i
1
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1
∑
h
2
=
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i
2
∑
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1
=
0
q
1
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i
1
∑
p
2
=
0
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2
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i
2
∑
m
1
=
0
p
1
+
p
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∑
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2
=
0
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−
i
1
−
p
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+
q
2
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i
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−
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2
(
S
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,
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,
m
1
(
σ
s
h
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,
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2
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(
m
1
+
1
)
!
S
s
q
1
−
i
1
−
p
1
,
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2
−
i
2
−
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2
,
m
2
(
σ
s
i
1
−
h
1
,
i
2
−
h
2
,
j
)
(
m
2
+
1
)
!
+
[
S
s
p
1
,
p
2
,
m
1
(
σ
s
i
1
−
h
1
,
i
2
−
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2
,
j
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,
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s
q
1
−
i
1
−
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1
,
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2
−
i
2
−
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2
,
m
2
(
σ
s
h
1
,
h
2
,
j
)
]
(
m
1
+
m
2
+
2
)
(
m
1
+
1
)
!
m
2
!
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{\displaystyle {\begin{aligned}Q_{s}^{q_{1},q_{2},j}:=\sum _{i_{1}=2}^{q_{1}}\sum _{i_{2}=0}^{q_{2}}\sum _{h_{1}=1}^{i_{1}-1}\sum _{h_{2}=0}^{i_{2}}&\sum _{p_{1}=0}^{q_{1}-i_{1}}\sum _{{p_{2}}=0}^{q_{2}-i_{2}}\ \sum _{m_{1}=0}^{p_{1}+p_{2}}\ \sum _{{m_{2}}=0}^{q_{1}-i_{1}-p_{1}+q_{2}-i_{2}-p_{2}}\\&{\Bigg (}{{\frac {S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}}{({m_{1}}+1)!}}{\frac {S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )}}{({m_{2}}+1)!}}}\\&\qquad \qquad +{\frac {{\big [}S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )},S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}{\big ]}}{({m_{1}}+{m_{2}}+2)({m_{1}}+1)!{m_{2}}!}}{\Bigg )},\end{aligned}}}
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{\displaystyle {\begin{aligned}S_{s}^{r-1,n-r,0}(A)&:={\begin{cases}A&{\text{if }}r=n=1,\\0&{\text{otherwise}},\end{cases}}\\S_{s}^{r-1,n-r,i}(A)&:=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots +k_{i}=n-r\end{array}}{\big [}Y_{s}^{(j_{1},k_{1})},{\big [}\dots ,{\big [}Y_{s}^{(j_{i},k_{i})},A_{s}{\big ]}\dots {\big ]}{\big ]}\\&=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots k_{i}=n-r\end{array}}\operatorname {ad} _{Y_{s}^{(j_{1},k_{1})}}\circ \cdots \circ \operatorname {ad} _{Y_{s}^{(j_{i},k_{i})}}(A_{s}),\qquad i\in \mathbb {N} .\end{aligned}}}
Magnus, W. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 1954, VII (4): 649–673. doi:10.1002/cpa.3160070404 Blanes, S.; Casas, F.; Oteo, J.A.; Ros, J. Magnus and Fer expansions for matrix differential equations: The convergence problem. J. Phys. A: Math. Gen. 1998, 31 (1): 259–268. Bibcode:1998JPhA...31..259B doi:10.1088/0305-4470/31/1/023 Iserles, A.; Nørsett, S. P. On the solution of linear differential equations in Lie groups. Phil. Trans. R. Soc. Lond. A. 1999, 357 (1754): 983–1019. Bibcode:1999RSPTA.357..983I CiteSeerX 10.1.1.15.4614 S2CID 90949835 doi:10.1098/rsta.1999.0362 Blanes, S.; Casas, F.; Oteo, J.A.; Ros, J. The Magnus expansion and some of its applications. Phys. Rep. 2009, 470 (5–6): 151–238. Bibcode:2009PhR...470..151B S2CID 115177329 arXiv:0810.5488 doi:10.1016/j.physrep.2008.11.001 Kamm, K.; Pagliarani, S.; Pascucci, A. On the Stochastic Magnus Expansion and Its Application to SPDEs. Journal of Scientific Computing. 2021, 89 (3): 56. S2CID 211259118 arXiv:2001.01098 doi:10.1007/s10915-021-01633-6
![{\displaystyle {\begin{aligned}\Omega _{1}(t)&=\int _{0}^{t}A(t_{1})\,dt_{1},\\\Omega _{2}(t)&={\frac {1}{2}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\,[A(t_{1}),A(t_{2})],\\\Omega _{3}(t)&={\frac {1}{6}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\,{\Bigl (}{\big [}A(t_{1}),[A(t_{2}),A(t_{3})]{\big ]}+{\big [}A(t_{3}),[A(t_{2}),A(t_{1})]{\big ]}{\Bigr )},\\\Omega _{4}(t)&={\frac {1}{12}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\int _{0}^{t_{3}}dt_{4}\,\left({\Big [}{\big [}[A_{1},A_{2}],A_{3}{\big ]},A_{4}{\Big ]}\right.\\&\qquad +{\Big [}A_{1},{\big [}[A_{2},A_{3}],A_{4}{\big ]}{\Big ]}+{\Big [}A_{1},{\big [}A_{2},[A_{3},A_{4}]{\big ]}{\Big ]}+\left.{\Big [}A_{2},{\big [}A_{3},[A_{4},A_{1}]{\big ]}{\Big ]}\right),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d98fd6d46f84435fb731a73803d8446226f27580)
![{\displaystyle S_{n}^{(j)}=\sum _{m=1}^{n-j}\left[\Omega _{m},S_{n-m}^{(j-1)}\right],\quad 2\leq j\leq n-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81703c24da029c6762a365928f7467ddc49d6777)
![{\displaystyle S_{n}^{(1)}=\left[\Omega _{n-1},A\right],\quad S_{n}^{(n-1)}=\operatorname {ad} _{\Omega _{1}}^{n-1}(A),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d35b59f701c4d7052c38bdb2b6dbceecb7b8b3bc)
![{\displaystyle \operatorname {ad} _{\Omega }^{0}A=A,\quad \operatorname {ad} _{\Omega }^{k+1}A=[\Omega ,\operatorname {ad} _{\Omega }^{k}A],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3579110e0417de1ffa276dda567ffaf7499ed3c6)
![{\textstyle \left(W_{t}\right)_{t\in [0,T]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c398893e410dc670983bfbbad4d2d322df05e191)
![{\displaystyle {\begin{aligned}Y_{t}^{(0,0)}&=0,\\Y_{t}^{(1,0)}&=\int _{0}^{t}A_{s}^{(j)}\,dW_{s}^{j},\\Y_{t}^{(0,1)}&=\int _{0}^{t}B_{s}\,ds,\\Y_{t}^{(2,0)}&=-{\frac {1}{2}}\int _{0}^{t}{\big (}A_{s}^{(j)}{\big )}^{2}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}A_{r}^{(i)}\,dW_{r}^{i}{\Big ]}dW_{s}^{j},\\Y_{t}^{(1,1)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}A_{r}^{(j)}\,dW_{r}{\Big ]}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}B_{r}\,dr{\Big ]}\,dW_{s}^{j},\\Y_{t}^{(0,2)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}B_{r}\,dr{\Big ]}\,ds.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed5c55f67f226450b8d429fe79b917589101af1)
![{\textstyle X=(X_{t})_{t\in [0,T]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eff1b6cd34a6413e006f83160616dc98472dab6)
![{\textstyle \|A_{\cdot }\|_{T}=\|\|A_{t}\|_{F}\|_{L^{\infty }(\Omega \times [0,T])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c21da16bf89b8a325a71ad502fb3ee0bc458f409)
![{\displaystyle \mathbb {P} (\tau \leq t)\leq Ct,\qquad t\in [0,T].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28da67281b17b15920260b463065d7e5b17ea431)
![{\displaystyle {\begin{aligned}Q_{s}^{q_{1},q_{2},j}:=\sum _{i_{1}=2}^{q_{1}}\sum _{i_{2}=0}^{q_{2}}\sum _{h_{1}=1}^{i_{1}-1}\sum _{h_{2}=0}^{i_{2}}&\sum _{p_{1}=0}^{q_{1}-i_{1}}\sum _{{p_{2}}=0}^{q_{2}-i_{2}}\ \sum _{m_{1}=0}^{p_{1}+p_{2}}\ \sum _{{m_{2}}=0}^{q_{1}-i_{1}-p_{1}+q_{2}-i_{2}-p_{2}}\\&{\Bigg (}{{\frac {S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}}{({m_{1}}+1)!}}{\frac {S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )}}{({m_{2}}+1)!}}}\\&\qquad \qquad +{\frac {{\big [}S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )},S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}{\big ]}}{({m_{1}}+{m_{2}}+2)({m_{1}}+1)!{m_{2}}!}}{\Bigg )},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1007971f2e29febe1a5af9f266bad2f0617ae6d8)
![{\displaystyle {\begin{aligned}S_{s}^{r-1,n-r,0}(A)&:={\begin{cases}A&{\text{if }}r=n=1,\\0&{\text{otherwise}},\end{cases}}\\S_{s}^{r-1,n-r,i}(A)&:=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots +k_{i}=n-r\end{array}}{\big [}Y_{s}^{(j_{1},k_{1})},{\big [}\dots ,{\big [}Y_{s}^{(j_{i},k_{i})},A_{s}{\big ]}\dots {\big ]}{\big ]}\\&=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots k_{i}=n-r\end{array}}\operatorname {ad} _{Y_{s}^{(j_{1},k_{1})}}\circ \cdots \circ \operatorname {ad} _{Y_{s}^{(j_{i},k_{i})}}(A_{s}),\qquad i\in \mathbb {N} .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30ca63471e2d7cd8ac3417c7f54a3e2df56bdfcc)
. S2CID 90949835. doi:10.1098/rsta.1999.0362.