Difference between revisions of "Multinomial theorem"
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From <math>\alpha \in {}^{(\nu)}\mathbb{C}, \binom{\alpha}{n}:=\widetilde{n!}\alpha\acute{\alpha}...(\grave{\alpha}-n)</math> and <math>\left|\binom{\alpha}{\grave{m}}/\binom{\alpha}{m}\right|<1</math> for all <math>m \ge \nu</math> and <math>\binom{\alpha}{0}:=1</math>, it follows for <math>z \in \mathbb{D}^\ll</math> or <math>z \in {}^{(\omega)}\mathbb{C}</math> for <math>\alpha \in {}^{(\omega)}\mathbb{N}</math> the [[w:Taylor series|<span class="wikipedia">Taylor series</span>]] centred on 0 by [[w:Differential calculus|<span class="wikipedia">derivating</span>]] | From <math>\alpha \in {}^{(\nu)}\mathbb{C}, \binom{\alpha}{n}:=\widetilde{n!}\alpha\acute{\alpha}...(\grave{\alpha}-n)</math> and <math>\left|\binom{\alpha}{\grave{m}}/\binom{\alpha}{m}\right|<1</math> for all <math>m \ge \nu</math> and <math>\binom{\alpha}{0}:=1</math>, it follows for <math>z \in \mathbb{D}^\ll</math> or <math>z \in {}^{(\omega)}\mathbb{C}</math> for <math>\alpha \in {}^{(\omega)}\mathbb{N}</math> the [[w:Taylor series|<span class="wikipedia">Taylor series</span>]] centred on 0 by [[w:Differential calculus|<span class="wikipedia">derivating</span>]] | ||
− | <div style="text-align:center;"><math>{\grave{z}}^\alpha= | + | <div style="text-align:center;"><math>{\grave{z}}^\alpha={+}_{n=0}^{\omega}{\binom{\alpha}{n}z^n}.\square</math></div> |
=== Multinomial theorem === | === Multinomial theorem === | ||
− | For <math>z, \ | + | For <math>z, \zeta \in {}^{(\omega)}\mathbb{C}^{k}, n^T \in {}^{(\omega)}\mathbb{N}^{k}, k, m \in {}^{\omega}\mathbb{N}^{*}, z^n := z_1^{n_1} ... z_k^{n_k}</math> and <math>\binom{m}{n} := \widetilde{n_1! ... {n}_k!}m!\;(k \ge 2)</math>, it holds that |
− | <div style="text-align:center;"><math>\left({\underline{1}}_k^Tz\right)^m= | + | <div style="text-align:center;"><math>\left({\underline{1}}_k^Tz\right)^m={+}_{n\underline{1}_k=m}{\binom{m}{n}z^n}.</math></div> |
==== Proof: ==== | ==== Proof: ==== | ||
Cases <math>k \in \{1, 2\}</math> are clear. [[w:Mathematical induction|<span class="wikipedia">Induction step</span>]] from <math>k</math> to <math>\grave{k}</math> by summarising the last two summands for <math>p \in {}^{\omega}\mathbb{N}</math>: | Cases <math>k \in \{1, 2\}</math> are clear. [[w:Mathematical induction|<span class="wikipedia">Induction step</span>]] from <math>k</math> to <math>\grave{k}</math> by summarising the last two summands for <math>p \in {}^{\omega}\mathbb{N}</math>: | ||
− | <div style="text-align:center;"><math>({(1,z^T){\underline{1}}_{\grave{k}})}^m= | + | <div style="text-align:center;"><math>({(1,z^T){\underline{1}}_{\grave{k}})}^m={+}_{{(n_0,n)\underline{1}}_{\grave{k}}=m}{\binom{m}{n_0,n}z^n}</math> because of <math>\binom{m}{n_1,...,n_{k-2},p} \binom{p}{n_{\grave{k}},n_k} = \binom{m}{n_1,...,n_k}.</math></div> |
Induction step from <math>m</math> to <math>\grave{m}</math> for <math>\grave{n} := n+(1,0, ... ,0)</math> by individual [[w:Integral|<span class="wikipedia">integration</span>]]: | Induction step from <math>m</math> to <math>\grave{m}</math> for <math>\grave{n} := n+(1,0, ... ,0)</math> by individual [[w:Integral|<span class="wikipedia">integration</span>]]: | ||
− | <div style="text-align:center;"><math>{\grave{m}\ | + | <div style="text-align:center;"><math>{\grave{m}{\uparrow}_{0}^{\zeta_1}{\left({\underline{1}}_k^Tz\right)^m{\downarrow}z_1}=\left.\left({\underline{1}}_k^T\zeta\right)^{\grave{m}}\right|_0^{{\zeta}_1}={+}_{{\grave{n}}{\underline{1}}_k=\grave{m}}\binom{\grave{m}}{\grave{n}}z^{\grave{n}}}.\square</math></div> |
=== General Leibniz formula=== | === General Leibniz formula=== | ||
− | For <math>\ | + | For <math>{\downarrow}^n := {\downarrow}_1^{n_1}...{\downarrow}_k^{n_k}</math> and <math>{\downarrow}_j^{n_j} := {\downarrow}^{n_j}/{\downarrow}{z_j}^{n_j}</math>, it follows for <math>m^T, n^T \in {}^{(\omega)}\mathbb{N}^{k}, j, k \in {}^{(\omega)}\mathbb{N}^*</math> and differentiable <math>f = f_1\cdot...\cdot f_k \in {}^{(\omega)}\mathbb{C}</math> from the multinomial theorem that |
− | <div style="text-align:center;"><math>\ | + | <div style="text-align:center;"><math>{\downarrow}^mf = {+}_{n\underline{1}_k=||m||_1}{\binom{||m||_1}{n}{\downarrow}^nf}.\square</math></div> |
== See also == | == See also == |
Revision as of 16:31, 25 July 2022
Theorem (binomial series)
From [math]\displaystyle{ \alpha \in {}^{(\nu)}\mathbb{C}, \binom{\alpha}{n}:=\widetilde{n!}\alpha\acute{\alpha}...(\grave{\alpha}-n) }[/math] and [math]\displaystyle{ \left|\binom{\alpha}{\grave{m}}/\binom{\alpha}{m}\right|<1 }[/math] for all [math]\displaystyle{ m \ge \nu }[/math] and [math]\displaystyle{ \binom{\alpha}{0}:=1 }[/math], it follows for [math]\displaystyle{ z \in \mathbb{D}^\ll }[/math] or [math]\displaystyle{ z \in {}^{(\omega)}\mathbb{C} }[/math] for [math]\displaystyle{ \alpha \in {}^{(\omega)}\mathbb{N} }[/math] the Taylor series centred on 0 by derivating
Multinomial theorem
For [math]\displaystyle{ z, \zeta \in {}^{(\omega)}\mathbb{C}^{k}, n^T \in {}^{(\omega)}\mathbb{N}^{k}, k, m \in {}^{\omega}\mathbb{N}^{*}, z^n := z_1^{n_1} ... z_k^{n_k} }[/math] and [math]\displaystyle{ \binom{m}{n} := \widetilde{n_1! ... {n}_k!}m!\;(k \ge 2) }[/math], it holds that
Proof:
Cases [math]\displaystyle{ k \in \{1, 2\} }[/math] are clear. Induction step from [math]\displaystyle{ k }[/math] to [math]\displaystyle{ \grave{k} }[/math] by summarising the last two summands for [math]\displaystyle{ p \in {}^{\omega}\mathbb{N} }[/math]:
Induction step from [math]\displaystyle{ m }[/math] to [math]\displaystyle{ \grave{m} }[/math] for [math]\displaystyle{ \grave{n} := n+(1,0, ... ,0) }[/math] by individual integration:
General Leibniz formula
For [math]\displaystyle{ {\downarrow}^n := {\downarrow}_1^{n_1}...{\downarrow}_k^{n_k} }[/math] and [math]\displaystyle{ {\downarrow}_j^{n_j} := {\downarrow}^{n_j}/{\downarrow}{z_j}^{n_j} }[/math], it follows for [math]\displaystyle{ m^T, n^T \in {}^{(\omega)}\mathbb{N}^{k}, j, k \in {}^{(\omega)}\mathbb{N}^* }[/math] and differentiable [math]\displaystyle{ f = f_1\cdot...\cdot f_k \in {}^{(\omega)}\mathbb{C} }[/math] from the multinomial theorem that