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Carleman matrix
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<h2 id="definition">Definition</h2> <p>The Carleman matrix of an infinitely differentiable function f ( x ) {\displaystyle f(x)} is defined as: </p> M [ f ] j k = 1 k ! [ d k d x k ( f ( x ) ) j ] x = 0 , {\displaystyle M[f]_{jk}={\frac {1}{k!}}\left[{\frac {d^{k}}{dx^{k}}}(f(x))^{j}\right]_{x=0}~,} <p>so as to satisfy the (<a href="/facts/Taylor_series/M0aTuftH">Taylor series</a>) equation: </p> ( f ( x ) ) j = ∑ k = 0 ∞ M [ f ] j k x k . {\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}.} <p>For instance, the computation of f ( x ) {\displaystyle f(x)} by </p> f ( x ) = ∑ k = 0 ∞ M [ f ] 1 , k x k . {\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.~} <p>simply amounts to the dot-product of row 1 of M [ f ] {\displaystyle M[f]} with a column vector [ 1 , x , x 2 , x 3 , . . . ] τ {\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }} . </p><p>The entries of M [ f ] {\displaystyle M[f]} in the next row give the 2nd power of f ( x ) {\displaystyle f(x)} : </p> f ( x ) 2 = ∑ k = 0 ∞ M [ f ] 2 , k x k , {\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}~,} <p>and also, in order to have the zeroth power of f ( x ) {\displaystyle f(x)} in M [ f ] {\displaystyle M[f]} , we adopt the row 0 containing zeros everywhere except the first position, such that </p> f ( x ) 0 = 1 = ∑ k = 0 ∞ M [ f ] 0 , k x k = 1 + ∑ k = 1 ∞ 0 ⋅ x k . {\displaystyle f(x)^{0}=1=\sum _{k=0}^{\infty }M[f]_{0,k}x^{k}=1+\sum _{k=1}^{\infty }0\cdot x^{k}~.} <p>Thus, the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> of M [ f ] {\displaystyle M[f]} with the column vector [ 1 , x , x 2 , . . . ] T {\displaystyle {\begin{bmatrix}1,x,x^{2},...\end{bmatrix}}^{T}} yields the column vector [ 1 , f ( x ) , f ( x ) 2 , . . . ] T {\displaystyle \left[1,f(x),f(x)^{2},...\right]^{T}} , i.e., </p> M [ f ] [ 1 x x 2 x 3 ⋮ ] = [ 1 f ( x ) ( f ( x ) ) 2 ( f ( x ) ) 3 ⋮ ] . {\displaystyle M[f]{\begin{bmatrix}1\\x\\x^{2}\\x^{3}\\\vdots \end{bmatrix}}={\begin{bmatrix}1\\f(x)\\(f(x))^{2}\\(f(x))^{3}\\\vdots \end{bmatrix}}.} <h2 id="generalization">Generalization</h2> <p>A generalization of the Carleman matrix of a function can be defined around any point, such as: </p> M [ f ] x 0 = M x [ x − x 0 ] M [ f ] M x [ x + x 0 ] {\displaystyle M[f]_{x_{0}}=M_{x}[x-x_{0}]M[f]M_{x}[x+x_{0}]} <p>or M [ f ] x 0 = M [ g ] {\displaystyle M[f]_{x_{0}}=M[g]} where g ( x ) = f ( x + x 0 ) − x 0 {\displaystyle g(x)=f(x+x_{0})-x_{0}} . This allows the <a href="/facts/Matrix_power/qa8FI4ko">matrix power</a> to be related as: </p> ( M [ f ] x 0 ) n = M x [ x − x 0 ] M [ f ] n M x [ x + x 0 ] {\displaystyle (M[f]_{x_{0}})^{n}=M_{x}[x-x_{0}]M[f]^{n}M_{x}[x+x_{0}]} <h3>General Series</h3> Another way to generalize it even further is think about a general series in the following way: Let h ( x ) = ∑ n c n ( h ) ⋅ ψ n ( x ) {\displaystyle h(x)=\sum _{n}c_{n}(h)\cdot \psi _{n}(x)} be a series approximation of h ( x ) {\displaystyle h(x)} , where { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} is a basis of the space containing h ( x ) {\displaystyle h(x)} Assuming that { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} is also a basis for f ( x ) {\displaystyle f(x)} , We can define G [ f ] m n = c n ( ψ m ∘ f ) {\displaystyle G[f]_{mn}=c_{n}(\psi _{m}\circ f)} , therefore we have ψ m ∘ f = ∑ n c n ( ψ m ∘ f ) ⋅ ψ n = ∑ n G [ f ] m n ⋅ ψ n {\displaystyle \psi _{m}\circ f=\sum _{n}c_{n}(\psi _{m}\circ f)\cdot \psi _{n}=\sum _{n}G[f]_{mn}\cdot \psi _{n}} , now we can prove that G [ g ∘ f ] = G [ g ] ⋅ G [ f ] {\displaystyle G[g\circ f]=G[g]\cdot G[f]} , if we assume that { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} is also a basis for g ( x ) {\displaystyle g(x)} and g ( f ( x ) ) {\displaystyle g(f(x))} . Let g ( x ) {\displaystyle g(x)} be such that ψ l ∘ g = ∑ m G [ g ] l m ⋅ ψ m {\displaystyle \psi _{l}\circ g=\sum _{m}G[g]_{lm}\cdot \psi _{m}} where G [ g ] l m = c m ( ψ l ∘ g ) {\displaystyle G[g]_{lm}=c_{m}(\psi _{l}\circ g)} . Now ∑ n G [ g ∘ f ] l n ψ n = ψ l ∘ ( g ∘ f ) = ( ψ l ∘ g ) ∘ f = ∑ m G [ g ] l m ( ψ m ∘ f ) = ∑ m G [ g ] l m ∑ n G [ f ] m n ψ n = ∑ n , m G [ g ] l m G [ f ] m n ψ n = ∑ n ( ∑ m G [ g ] l m G [ f ] m n ) ψ n {\displaystyle {\begin{aligned}\sum _{n}G[g\circ f]_{ln}\psi _{n}=\psi _{l}\circ (g\circ f)&=(\psi _{l}\circ g)\circ f\\&=\sum _{m}G[g]_{lm}(\psi _{m}\circ f)\\&=\sum _{m}G[g]_{lm}\sum _{n}G[f]_{mn}\psi _{n}\\&=\sum _{n,m}G[g]_{lm}G[f]_{mn}\psi _{n}\\&=\sum _{n}(\sum _{m}G[g]_{lm}G[f]_{mn})\psi _{n}\end{aligned}}} Comparing the first and the last term, and from { ψ n ( x ) } n {\displaystyle \{\psi _{n}(x)\}_{n}} being a base for f ( x ) {\displaystyle f(x)} , g ( x ) {\displaystyle g(x)} and g ( f ( x ) ) {\displaystyle g(f(x))} it follows that G [ g ∘ f ] = ∑ m G [ g ] l m G [ f ] m n = G [ g ] ⋅ G [ f ] {\displaystyle G[g\circ f]=\sum _{m}G[g]_{lm}G[f]_{mn}=G[g]\cdot G[f]} <h4>Examples</h4> <h5>Rederive (Taylor) Carleman Matrix</h5> <p>If we set ψ n ( x ) = x n {\displaystyle \psi _{n}(x)=x^{n}} we have the Carleman matrix. Because h ( x ) = ∑ n c n ( h ) ⋅ ψ n ( x ) = ∑ n c n ( h ) ⋅ x n {\displaystyle h(x)=\sum _{n}c_{n}(h)\cdot \psi _{n}(x)=\sum _{n}c_{n}(h)\cdot x^{n}} then we know that the n-th coefficient c n ( h ) {\displaystyle c_{n}(h)} must be the nth-coefficient of the <a href="/facts/Taylor_series/M0aTuftH">taylor series</a> of h {\displaystyle h} . Therefore c n ( h ) = 1 n ! h ( n ) ( 0 ) {\displaystyle c_{n}(h)={\frac {1}{n!}}h^{(n)}(0)} Therefore G [ f ] m n = c n ( ψ m ∘ f ) = c n ( f ( x ) m ) = 1 n ! [ d n d x n ( f ( x ) ) m ] x = 0 {\displaystyle G[f]_{mn}=c_{n}(\psi _{m}\circ f)=c_{n}(f(x)^{m})={\frac {1}{n!}}\left[{\frac {d^{n}}{dx^{n}}}(f(x))^{m}\right]_{x=0}} Which is the Carleman matrix given above. <i>(It's important to note that this is not an orthornormal basis)</i> </p> <h5>Carleman Matrix For Orthonormal Basis</h5> <p>If { e n ( x ) } n {\displaystyle \{e_{n}(x)\}_{n}} is an orthonormal basis for a Hilbert Space with a defined inner product ⟨ f , g ⟩ {\displaystyle \langle f,g\rangle } , we can set ψ n = e n {\displaystyle \psi _{n}=e_{n}} and c n ( h ) {\displaystyle c_{n}(h)} will be ⟨ h , e n ⟩ {\displaystyle {\displaystyle \langle h,e_{n}\rangle }} . Then G [ f ] m n = c n ( e m ∘ f ) = ⟨ e m ∘ f , e n ⟩ {\displaystyle G[f]_{mn}=c_{n}(e_{m}\circ f)=\langle e_{m}\circ f,e_{n}\rangle } . </p> <h5>Carleman Matrix for Fourier Series</h5> <p>If e n ( x ) = e i n x {\displaystyle e_{n}(x)=e^{inx}} we have the analogous for <a href="/facts/Fourier_Series/s3fsxPAT">Fourier Series</a>. Let c ^ n {\displaystyle {\hat {c}}_{n}} and G ^ {\displaystyle {\hat {G}}} represent the carleman coefficient and matrix in the fourier basis. Because the basis is orthogonal, we have. </p> c ^ n ( h ) = ⟨ h , e n ⟩ = 1 2 π ∫ − π π h ( x ) ⋅ e − i n x d x {\displaystyle {\hat {c}}_{n}(h)=\langle h,e_{n}\rangle ={\cfrac {1}{2\pi }}\int _{-\pi }^{\pi }\displaystyle h(x)\cdot e^{-inx}dx} . <p>Then, therefore, G ^ [ f ] m n = c n ^ ( e m ∘ f ) = ⟨ e m ∘ f , e n ⟩ {\displaystyle {\hat {G}}[f]_{mn}={\hat {c_{n}}}(e_{m}\circ f)=\langle e_{m}\circ f,e_{n}\rangle } which is </p> G ^ [ f ] m n = 1 2 π ∫ − π π e i m f ( x ) ⋅ e − i n x d x {\displaystyle {\hat {G}}[f]_{mn}={\cfrac {1}{2\pi }}\int _{-\pi }^{\pi }\displaystyle e^{imf(x)}\cdot e^{-inx}dx} <h2 id="properties">Properties</h2> <p>Carleman matrices satisfy the fundamental relationship </p> <ul><li> M [ f ∘ g ] = M [ f ] M [ g ] , {\displaystyle M[f\circ g]=M[f]M[g]~,} </li></ul> <p>which makes the Carleman matrix <i>M</i> a (direct) representation of f ( x ) {\displaystyle f(x)} . Here the term f ∘ g {\displaystyle f\circ g} denotes the composition of functions f ( g ( x ) ) {\displaystyle f(g(x))} . </p><p>Other properties include: </p> <ul><li> M [ f n ] = M [ f ] n {\displaystyle \,M[f^{n}]=M[f]^{n}} , where f n {\displaystyle \,f^{n}} is an <a href="/facts/Iterated_function/Vzsx64An">iterated function</a> and</li> <li> M [ f − 1 ] = M [ f ] − 1 {\displaystyle \,M[f^{-1}]=M[f]^{-1}} , where f − 1 {\displaystyle \,f^{-1}} is the <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a> (if the Carleman matrix is <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a>).</li></ul> <h2 id="examples">Examples</h2> <p>The Carleman matrix of a constant is: </p> M [ a ] = ( 1 0 0 ⋯ a 0 0 ⋯ a 2 0 0 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[a]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&0&0&\cdots \\a^{2}&0&0&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} <p>The Carleman matrix of the identity function is: </p> M x [ x ] = ( 1 0 0 ⋯ 0 1 0 ⋯ 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[x]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&1&0&\cdots \\0&0&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} <p>The Carleman matrix of a constant addition is: </p> M x [ a + x ] = ( 1 0 0 ⋯ a 1 0 ⋯ a 2 2 a 1 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[a+x]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&1&0&\cdots \\a^{2}&2a&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} <p>The Carleman matrix of the <a href="/facts/Successor_function/R1hhkSEG">successor function</a> is equivalent to the <a href="/facts/Binomial_coefficient/Tsx7eY5h">Binomial coefficient</a>: </p> M x [ 1 + x ] = ( 1 0 0 0 ⋯ 1 1 0 0 ⋯ 1 2 1 0 ⋯ 1 3 3 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[1+x]=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&1&0&0&\cdots \\1&2&1&0&\cdots \\1&3&3&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ 1 + x ] j k = ( j k ) {\displaystyle M_{x}[1+x]_{jk}={\binom {j}{k}}} <p>The Carleman matrix of the <a href="/facts/Logarithm/QeXaV97q">logarithm</a> is related to the (signed) <a href="/facts/Stirling_numbers_of_the_first_kind/Twdk4kA5">Stirling numbers of the first kind</a> scaled by <a href="/facts/Factorial/kkfy1Bwe">factorials</a>: </p> M x [ log ( 1 + x ) ] = ( 1 0 0 0 0 ⋯ 0 1 − 1 2 1 3 − 1 4 ⋯ 0 0 1 − 1 11 12 ⋯ 0 0 0 1 − 3 2 ⋯ 0 0 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[\log(1+x)]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&-{\frac {1}{2}}&{\frac {1}{3}}&-{\frac {1}{4}}&\cdots \\0&0&1&-1&{\frac {11}{12}}&\cdots \\0&0&0&1&-{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ log ( 1 + x ) ] j k = s ( k , j ) j ! k ! {\displaystyle M_{x}[\log(1+x)]_{jk}=s(k,j){\frac {j!}{k!}}} <p>The Carleman matrix of the <a href="/facts/Logarithm/QeXaV97q">logarithm</a> is related to the (unsigned) <a href="/facts/Stirling_numbers_of_the_first_kind/Twdk4kA5">Stirling numbers of the first kind</a> scaled by <a href="/facts/Factorial/kkfy1Bwe">factorials</a>: </p> M x [ − log ( 1 − x ) ] = ( 1 0 0 0 0 ⋯ 0 1 1 2 1 3 1 4 ⋯ 0 0 1 1 11 12 ⋯ 0 0 0 1 3 2 ⋯ 0 0 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[-\log(1-x)]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&\cdots \\0&0&1&1&{\frac {11}{12}}&\cdots \\0&0&0&1&{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ − log ( 1 − x ) ] j k = | s ( k , j ) | j ! k ! {\displaystyle M_{x}[-\log(1-x)]_{jk}=|s(k,j)|{\frac {j!}{k!}}} <p>The Carleman matrix of the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> is related to the <a href="/facts/Stirling_numbers_of_the_second_kind/qg5UrHmk">Stirling numbers of the second kind</a> scaled by <a href="/facts/Factorial/kkfy1Bwe">factorials</a>: </p> M x [ exp ( x ) − 1 ] = ( 1 0 0 0 0 ⋯ 0 1 1 2 1 6 1 24 ⋯ 0 0 1 1 7 12 ⋯ 0 0 0 1 3 2 ⋯ 0 0 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[\exp(x)-1]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&{\frac {1}{2}}&{\frac {1}{6}}&{\frac {1}{24}}&\cdots \\0&0&1&1&{\frac {7}{12}}&\cdots \\0&0&0&1&{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ exp ( x ) − 1 ] j k = S ( k , j ) j ! k ! {\displaystyle M_{x}[\exp(x)-1]_{jk}=S(k,j){\frac {j!}{k!}}} <p>The Carleman matrix of <a href="/facts/Exponential_function/ewlYxvR2">exponential functions</a> is: </p> M x [ exp ( a x ) ] = ( 1 0 0 0 ⋯ 1 a a 2 2 a 3 6 ⋯ 1 2 a 2 a 2 4 a 3 3 ⋯ 1 3 a 9 a 2 2 9 a 3 2 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[\exp(ax)]=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&a&{\frac {a^{2}}{2}}&{\frac {a^{3}}{6}}&\cdots \\1&2a&2a^{2}&{\frac {4a^{3}}{3}}&\cdots \\1&3a&{\frac {9a^{2}}{2}}&{\frac {9a^{3}}{2}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)} M x [ exp ( a x ) ] j k = ( j a ) k k ! {\displaystyle M_{x}[\exp(ax)]_{jk}={\frac {(ja)^{k}}{k!}}} <p>The Carleman matrix of a constant multiple is: </p> M x [ c x ] = ( 1 0 0 ⋯ 0 c 0 ⋯ 0 0 c 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&c&0&\cdots \\0&0&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} <p>The Carleman matrix of a linear function is: </p> M x [ a + c x ] = ( 1 0 0 ⋯ a c 0 ⋯ a 2 2 a c c 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[a+cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&c&0&\cdots \\a^{2}&2ac&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} <p>The Carleman matrix of a function f ( x ) = ∑ k = 1 ∞ f k x k {\displaystyle f(x)=\sum _{k=1}^{\infty }f_{k}x^{k}} is: </p> M [ f ] = ( 1 0 0 ⋯ 0 f 1 f 2 ⋯ 0 0 f 1 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&f_{1}&f_{2}&\cdots \\0&0&f_{1}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} <p>The Carleman matrix of a function f ( x ) = ∑ k = 0 ∞ f k x k {\displaystyle f(x)=\sum _{k=0}^{\infty }f_{k}x^{k}} is: </p> M [ f ] = ( 1 0 0 ⋯ f 0 f 1 f 2 ⋯ f 0 2 2 f 0 f 1 f 1 2 + 2 f 0 f 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\f_{0}&f_{1}&f_{2}&\cdots \\f_{0}^{2}&2f_{0}f_{1}&f_{1}^{2}+2f_{0}f_{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} <h2 id="related-matrices">Related matrices</h2> <p>The Bell matrix or the Jabotinsky matrix of a function f ( x ) {\displaystyle f(x)} is defined as<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> B [ f ] j k = 1 j ! [ d j d x j ( f ( x ) ) k ] x = 0 , {\displaystyle B[f]_{jk}={\frac {1}{j!}}\left[{\frac {d^{j}}{dx^{j}}}(f(x))^{k}\right]_{x=0}~,} <p>so as to satisfy the equation </p> ( f ( x ) ) k = ∑ j = 0 ∞ B [ f ] j k x j , {\displaystyle (f(x))^{k}=\sum _{j=0}^{\infty }B[f]_{jk}x^{j}~,} <p>These matrices were developed in 1947 by Eri Jabotinsky to represent convolutions of polynomials.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> It is the <a href="/facts/Transpose/8wmsagGS">transpose</a> of the Carleman matrix and satisfy </p><p> B [ f ∘ g ] = B [ g ] B [ f ] , {\displaystyle B[f\circ g]=B[g]B[f]~,} which makes the Bell matrix <i>B</i> an <i>anti-representation</i> of f ( x ) {\displaystyle f(x)} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Carleman_linearization/yM49NfRo">Carleman linearization</a></li> <li><a href="/facts/Composition_operator/j2Tzx5w4">Composition operator</a></li> <li><a href="/facts/Function_composition/r5Pvnmth">Function composition</a></li> <li><a href="/facts/Schr%C3%B6der%27s_equation/jWWTbrra">Schröder's equation</a></li> <li><a href="/facts/Bell_polynomials/qQ9rxxbA">Bell polynomials</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>R Aldrovandi, <a href="https://web.archive.org/web/20120519152123/http://www.worldscibooks.com/physics/4772.html">Special Matrices of Mathematical Physics</a>: Stochastic, Circulant and Bell Matrices, World Scientific, 2001. (<a href="https://books.google.com/books?hl=en&lr=&id=wb9aLGfVsOwC">preview</a>)</li> <li>R. Aldrovandi, L. P. Freitas, Continuous Iteration of Dynamical Maps, online preprint, 1997.</li> <li>P. Gralewicz, K. Kowalski, Continuous time evolution from iterated maps and Carleman linearization, online preprint, 2000.</li> <li>K Kowalski and W-H Steeb, <a href="https://web.archive.org/web/20120208174226/http://www.worldscibooks.com/mathematics/1347.html">Nonlinear Dynamical Systems and Carleman Linearization</a>, World Scientific, 1991. (<a href="https://books.google.com/books?id=PTTCxQwFtMEC">preview</a>)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Knuth, D. (1992). "Convolution Polynomials". The Mathematica Journal. 2 (4): 67–78. arXiv:math/9207221. Bibcode:1992math......7221K. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Jabotinsky, Eri (1953). "Representation of functions by matrices. Application to Faber polynomials". Proceedings of the American Mathematical Society. 4 (4): 546–553. doi:10.1090/S0002-9939-1953-0059359-0. ISSN 0002-9939. <a href="https://www.ams.org/proc/1953-004-04/S0002-9939-1953-0059359-0/" target="_blank">https://www.ams.org/proc/1953-004-04/S0002-9939-1953-0059359-0/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lang, W. (2000). "On generalizations of the stirling number triangles". Journal of Integer Sequences. 3 (2.4): 1–19. Bibcode:2000JIntS...3...24L. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Jabotinsky, Eri (1947). "Sur la représentation de la composition de fonctions par un produit de matrices. Applicaton à l'itération de e^x et de e^x-1". Comptes rendus de l'Académie des Sciences. 224: 323–324. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>