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Regular aggregation sequences and their (in)finite series

REF. DATE:11 October 2015 CREATED/MODIFIED: 13 October 2015 28 views One Comment

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.

In the comments of this post, you see the precise notions and a preliminary result, which is quite open for discussion…

One Comment »

  • E. Rocha (author) said:

    NOTATION. I use < -1,+1 > 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 < 1,2,3,\dots > (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\}.

    Let n\in\bkN, v\in\bkR, and k\in\bkZ. Define the conditional function cf_n:\bkZ\times\bkR\rightarrow\bkR as

    (1)   \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 \left\{v|k\right\}_{n}.

    DEFINITION. A regular aggregation sequence (r.a.s.) is a sequence u\in\bkR^{\infty} such that exist fixed tupples R_u=< f_0,\dots,f_{|R_u|-1}> and \Gamma_u=< \alpha, \beta>, with f_j:\bkZ\times\bkZ\rightarrow\bkR, \alpha,\beta\in\bkZ and \alpha\neq 0, such that

    (2)   \begin{align*} u_i &= \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*}

    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 \mathcal{RS}.

    DEFINITION. The aggregation operation \varphi_{m,q}:\mathcal{RS}\rightarrow\mathcal{RS}, with m,q\in\bkNz and q<m, is the map such that,

        \[w = \varphi_{m,q}(u)\:\:\:\mbox{ and }\:\:\: w_i = \sum_{k=0}^{m-1} u_{mi-q+k}\:\:\mbox{for all } i\in\bkN.\]

    LEMMA. For m,m_1,m_2,q,q_1,q_2\in\bkNz and u\in\mathcal{RS}, the map \varphi_{m,q} has the following properties:

    (1) It is well-defined, i.e. \varphi_{m,q}(u)\in\mathcal{RS}.

    (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.

    (3) Fix u\in\RS and define \Phi_u:\bkN\rightarrow\mathcal{RS} by

        \[\Phi_u(m)=\varphi_{m,0}(u).\]

    The function \Phi_u is an homomorphism.

    (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}.

    (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

        \[\varphi_{m,0}(u)=\varphi_{p_k,0}^{n_k}\cdots\varphi_{p_1,0}^{n_1}(u).\]

    See the full content in this Mathdir topic.

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