Regular aggregation sequences and their (in)finite series

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  • #685
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    E. Rocha
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    Motivated by the (renormalization) of some classical divergent series in String Theory, e.g.

     \sum_{n=1}^{+\infty} n = -\frac{1}{12} \:\:\mbox{ (yes, it means) } 1 + 2+ 3+ 4+5+6+\dots = -\frac{1}{12}

    (see here), I start thinking on the convergence meaning of a class of divergent series, trying to make sense of them without following some of the (standard) approaches as Hardy resummation or Zeta function regularization (e.g. wikipedia). The main idea was to see if it is possible to define a class of sequences and a equivalence operation, by aggregating terms, such that their “value” is determined by some constant sequence in the same equivalent class.

         \section*{Basic definitions} \textbf{Notation.} I use $\ls -1,+1 \rs$ to denote a tupple (e.g. a 2-tupple), of real numbers, since $\{-1,+1\}$ means a set (i.e. no order and no element repetition) and $(-1,+1)$ is an open interval. We will also consider tupples of functions. The size of a tupple $u$ is denoted by $|u|$. Any sequence $\ls -1,+1, \dots \rs$ (an infinite-tupple) can be extended by zeros on the left, so it is infinite in the two directions. Therefore the index of a sequence is an element of $\bkZ$, where the first element of a sequence $u$ is denoted by $u_0$. Moreover, since $u_{-i}=0$ for all $i\in\bkN$, the relevant set of indices is $\bkNz$. For $m\in\bkN$, we denote by $\bkN_m$ the integers that are congruent to $m$, i.e. the set $\{0,1,\dots,m-1\}$. \begin{dfn} Let $n\in\bkN$, $v\in\bkR$, and $k\in\bkZ$. Define the \textbf{conditional function} $\cf_n:\bkZ\times\bkR\rightarrow\bkR$ as \begin{align*} \cf_n(v,k) := \left\{\begin{array}{ll} v & \mbox{ if } k=0 \mbox{ (mod n)},\\ 0 & \mbox{ otherwise}, \end{array} \right. \end{align*} which, for convenience, we just write as $\cfn{n}{v}{k}.$ \end{dfn} \begin{dfn} A \textbf{regular aggregation sequence} (r.a.s.) is a sequence $u\in\bkR^{\infty}$ such that \begin{enumerate} \item[(i)] Exist fixed tupples $R_u=\ls f_0,\dots,f_{|R_u|-1}\rs$ and $\Gamma_u=\ls \alpha, \beta\rs$, with $f_j:\bkZ\times\bkZ\rightarrow\bkR$, $\alpha,\beta\in\bkZ$ and $\alpha\neq 0$, such that \begin{align*} u_i %%% &= \sum_{j=0}^{|R_u|-1} \cf_{|R_u|}(i,j,f_j,\Gamma)\\ &= \sum_{j=0}^{|R_u|-1} \cfn{|R_u|}{f_j(\alpha i+\beta,j)}{\alpha i+\beta-j} \:\:\:\mbox{ for all }i\in\bkN. \end{align*} \end{enumerate} We call the pair $(R_u,\Gamma_u)$ a realization of u, which is not unique for each sequence u. The set of all r.a.s. is denoted by $\RS$. \end{dfn} \begin{lmm} Let $u\in\RS$ with realization $R_u=\ls f_0,\dots,f_{|R_u|-1}\rs$ and $\Gamma_u=\ls \alpha,\beta\rs$. Given $m\in\bkN\backslash\{1\}$, the following is also a realization $$ \tilde{R}_u=\ls f_0,\dots,f_{|R_u|-1},\cdots,f_0,\dots,f_{|R_u|-1}\rs\:\:\mbox{ and }\:\:\tilde{\Gamma}_u=\Gamma_u,$$ where $|\tilde{R}_u|=m|R_u|$. This is called a $m$-copy of $R_u$ and we denoted it by $(R_u,\Gamma_u)^m$. \end{lmm}

  • #686
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    E. Rocha
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         \section*{Simple Examples} \textbf{Example.} For the sequence $u=\ls 1,2,3,4,\dots\rs$ with realization $(R,\Gamma)=(\ls f \rs, \ls 1,0\rs)$ with $f(i,j)=i+1$, we have $u\in\RS$ because \begin{eqnarray*} u_i &=& \cfn{1}{i+1}{i} = i+1, \end{eqnarray*} for any $i\in\bkNz$. Note that $(R,\Gamma)=(\ls Id_i, Id_i \rs,\ls 1,1\rs)$ is also a realization because \begin{eqnarray*} u_i &=& \cfn{2}{i+1}{i+1}+ \cfn{2}{i+1}{i} = i+1, \end{eqnarray*} for any $i\in\bkNz$. In what follows, we may abuse the notation writing $R=\ls i+1\rs$ instead of $R=\ls f \rs$ with $f(i,j)=i+1$.\\ \textbf{Example.} For the sequence $u=\ls+1,-1,+1,-1,\dots\rs$ with realization $(R,\Gamma)=(\ls +1,-1\rs,\ls 1,0\rs)$, we have $u\in\RS$ because \begin{eqnarray*} u_i &=& \cfn{2}{+1}{i}+\cfn{2}{-1}{i-1} = (-1)^i, \end{eqnarray*} for any $i\in\bkNz$.\\ \textbf{Example.} For the sequence $u=\ls +1,-2,+3,-4,\dots\rs$ with realization $(R,\Gamma)=(\ls (-1)^{i}(i+1)\rs,\ls 1,0\rs)$, we have $u\in\RS$ because \begin{eqnarray*} u_i &=& \cfn{1}{(-1)^{i}(i+1)}{i} = (-1)^i(i+1), \end{eqnarray*} for any $i\in\bkNz$. Consider now the realization $$(R,\Gamma)^2=(\ls (-1)^{i}(i+1), (-1)^{i}(i+1)\rs,\ls 1,0\rs),$$ we have \begin{eqnarray*} u_i %%% &=& \sum_{j=0}^{|R_u|-1} \cf_{|R_u|}(i,j,f_j,\Gamma)\\ &=& \cfn{2}{(-1)^{i}(i+1)}{i}+\cfn{2}{(-1)^{i}(i+1)}{i-1} = (-1)^i(i+1), \end{eqnarray*} for any $i\in\bkNz$.\\ \textbf{Example.} For the sequence $u=\ls +1,-1,-1,+2,+1,-3,-1,+4,\dots\rs$ with realization $(R,\Gamma)=\left(\ls (-1)^{\frac{i}{2}},(-1)^{\frac{i+1}{2}}\frac{i+1}{2},\rs,\ls 1,0\rs\right)$, we have $u\in\RS$ because \begin{align*} u_i &= \sum_{j=0}^{|R_u|-1} \cfn{|R_u|}{f_j(i,j)}{i-j} = \cfn{2}{(-1)^{\frac{i}{2}}}{i}+\cfn{2}{(-1)^{\frac{i+1}{2}}\frac{i+1}{2}}{i-1} \end{align*} for any $i\in\bkNz$.

  • #687
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    E. Rocha
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         \section*{Operations} The \textbf{aggregation operation} $\varphi_{m,q}:\mathcal{RS}\rightarrow\mathcal{RS}$ with $m,q\in\bkNz$ and $q<m$, is the map such that, $$ w = \varphi_{m,q}(u)\:\:\:\mbox{ and }\:\:\: w_i = \sum_{k=0}^{m-1} u_{mi-q+k}\:\:\mbox{for all } i\in\bkN.$$ \textbf{Example.} Consider the sequence $u=\ls +1,-2,+3,-4,\dots\rs$ with realization $(R,\Gamma)=(\ls (-1)^{i}(i+1)\rs,\ls 1,0\rs)$. For $w=\varphi_{2,0}(u)$, we have \begin{eqnarray} w_i &=& \sum_{k=0}^{1} u_{2i+k} = \sum_{k=0}^{1} \left(\sum_{j=0}^{0} \cfn{1}{f_j(2i+k,j)}{2i+k-j}\right)\\ &=& f_0(2i,0)+f_0(2i+1,1) = (-1)^{2i}(2i+1)+(-1)^{2i+1}(2i+2)=-1 \end{eqnarray*} For $w=\varphi_{2,1}(u)$, we have \begin{eqnarray} w_0 &=& u_{-1}+u_{1} = 1,\\ w_i &=& \sum_{k=0}^{1} u_{2i-1+k} = \sum_{k=0}^{1} \left(\sum_{j=0}^{0} \cfn{1}{f_j(2i-1+k,j)}{2i-1+k-j}\right)\\ &=& f_0(2i-1,0)+f_0(2i,1) = (-1)^{2i-1}2i+(-1)^{2i}(2i+1)=+1 \end{eqnarray} \textbf{Example.} Consider the sequence $u=\ls +1,-1,-1,+2,+1,-3,-1,+4,\dots\rs$ with realization $(R,\Gamma)=\left(\ls (-1)^{\frac{i}{2}},(-1)^{\frac{i+1}{2}}\frac{i+1}{2},\rs,\ls 1,0\rs\right)$. For $w=\varphi_{2,0}(u)$, we have \begin{eqnarray} w_i & = & \sum_{k=0}^{m-1} u_{mi-q+k} = u_{2i}+u_{2i+1}\\ &=& \sum_{k=0}^{m-1} \left(\sum_{j=0}^{|R_u|-1} \cfn{|R_u|}{f_j(\alpha(mi-q+k)+\beta,j)}{\alpha(mi-q+k)+\beta-j}\right) \\ &=& \sum_{k=0}^{1} \left(\sum_{j=0}^{1} \cfn{2}{f_j(2i+k,j)}{2i+k-j}\right) \\ &=& \cfn{2}{f_0(2i,0)}{2i}+\cfn{2}{f_1(2i,1)}{2i-1}+\cfn{2}{f_0(2i+1,0)}{2i+1} \\ &\& +\cfn{2}{f_1(2i+1,1)}{2i} \\ &=& \cfn{2}{f_0(2i,0)+f_1(2i+1,1)}{2i}+\cfn{2}{f_0(2i+1,0)+f_1(2i,1)}{2i+1} \\ &=& \cfn{2}{(-1)^{i}+(-1)^{i+1}(i+1)}{2i}+\cfn{2}{(-1)^{\frac{2i+1}{2}}\frac{2i+3}{2}}{2i+1} \\ &=& (-1)^{i+1}i = \ls 0,+1,-2,+3,-4,\dots\rs. \end{eqnarray} For $w=\varphi_{2,1}(u)$, we have \begin{eqnarray} w_0 & =& u_{-1}+u_{0} = +1 \\ w_i & = &\sum_{k=0}^{m-1} u_{mi-q+k} = u_{2i-1}+u_{2i}\\ &=& \sum_{k=0}^{m-1} \left(\sum_{j=0}^{|R_u|-1} \cfn{|R_u|}{f_j(\alpha(mi-q+k)+\beta,j)}{\alpha(mi-q+k)+\beta-j}\right) \\ &= &\sum_{k=0}^{1} \left(\sum_{j=0}^{1} \cfn{2}{f_j(2i-1+k,j)}{2i-1+k-j}\right) \\ &= &\cfn{2}{f_0(2i-1,0)}{2i-1}+\cfn{2}{f_1(2i-1,1)}{2i-2}+\cfn{2}{f_0(2i,0)}{2i} \\ &\ &+\cfn{2}{f_1(2i,1)}{2i-1} \\ &= \cfn{2}{f_0(2i-1,0)+f_1(2i,1)}{2i-1}+\cfn{2}{f_0(2i,0)+f_1(2i-1,1)}{2i} \\ &= &\cfn{2}{(-1)^{i}+(-1)^{i}i}{2i} = (-1)^{i+1}(i+1) = \ls +1,+2,-3,+4,\dots\rs. \end{eqnarray}

    • This reply was modified 1 year, 11 months ago by Profile photo of E. Rocha E. Rocha.
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  • #688
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    E. Rocha
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         \section*{A Lemma} We can show the following result which opens the use of Category theory. \begin{lmm}\label{lmm2} For $m,m_1,m_2,q,q_1,q_2\in\bkNz$ and $u\in\RS$, the map $\varphi_{m,q}$ has the following properties: \begin{enumerate} \item[(1)] It is well-defined, i.e. $\varphi_{m,q}(u)\in\RS$. \item[(2)] We have $$\varphi_{m_2,q_2}\varphi_{m_1,q_1}(u)=\varphi_{m_1m_2,\bar{q}}(u),$$ where $\bar{q}=\psi_{m_1,q_1}\psi_{m_2,q_2}(0)$, $\psi_{m,q}(\bar{q}):=\psi_{m}(q,\bar{q})$, and $$\psi_{m}:\bkN_{m}\times\bkN_{\bar{m}}\rightarrow\bkN_{m\bar{m}} : (q,\bar{q})\mapsto m\bar{q}+q$$ is a bijection. \item[(3)] Fix $u\in\RS$ and define $\Phi_u:\bkN\rightarrow\RS$ by $$ \Phi_u(m)=\varphi_{m,0}(u).$$ The function $\Phi_u$ is an homomorphism. \item[(4)] Let $n\in\bkN\backslash\{1\}$, $M=\{m_1,\dots,m_n\}$ and $Q=\{q_1,\dots,q_n\}$, then $$\varphi_{m_{n},q_{n}}\cdots\varphi_{m_1,q_1}(u)=\varphi_{m_1\cdots m_n,\bar{q}}(u),$$ where $\bar{q}=\psi_{m_{1},q_{1}}\cdots\psi_{m_{n},q_{n}}(0)\in\bkN_{m_1\cdots m_n}$. \item[(5)] We have $\varphi_{m_2,0}\varphi_{m_1,0}(u)=\varphi_{m_1,0}\varphi_{m_2,0}(u)$. Moreover, if $m=p_1^{n_1}\cdots p_k^{n_k}$ is a decomposition into prime numbers, then $$\varphi_{m,0}(u)=\varphi_{p_k,0}^{n_k}\cdots\varphi_{p_1,0}^{n_1}(u).$$ \end{enumerate} \end{lmm}

    Further can be explored if someone is interested in looking to it (please leave a reply). I did not saw if this definition relates with, for example, Dedekind sums.

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