Let $b>1$ be a fixed integer and $x_n$ , $y_n$ the $b$-digital sequences of positive real numbers $x$, $y$ (to the base $b$), i.e., $0\leq x_n, y_n<b$, there exists an integer $N$ such that $x_n=0$ (resp. $y_n=0$), for all $n>N$, and \begin{array}{c} x=\sum_{+\infty}^{-\infty}x_nb^n=\cdots x_{n+1} x_nx_{n-1}\cdots_b \\ \\ y=\sum_{+\infty}^{-\infty}y_nb^n=\cdots y_{n+1}y_ny_{n-1}\cdots _b \end{array}
Now, if $z=x+y$ and $z_n$ is the unique $b$-digital sequence corresponded to $z$, then
(a) Can one state a formula for evaluating the (two-sided) sequence $z_n$ according to $x_n$ , $y_n$?
(b) Are there any recurrence equations (formulas) for them?
(c) Does somebody know any references for it?
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Note that the usual algorithm of addition of two real numbers says: \begin{equation} \begin{array}{cccccc} \cdots & 0\; \mbox{or}\; 1 & 0\; \mbox{or}\; 1 & 0\; \mbox{or}\; 1 & \cdots \\ \cdots & x_{n+1} & x_{n} & x_{n-1} & \cdots \\ \cdots & y_{n+1} & y_{n} & y_{n-1} & \cdots & +\\ -- & -- & -- & -- & --\\ \cdots & z_{n+1} & z_{n}& z_{n-1} & \cdots \end{array} \end{equation} hence $z_n=x_n+y_n$ or $z_n=x_n+y_n+1$ mod $b$, but we do not know which one is correct, for a given integer $n$.
Refers to pre-babylonian carry rule. Start with a large enough $n$, $carry=0$, and proceed backward: if $x_n+y_n+carry<b$ then $z_n=x_n+y_n+carry$ and $carry=0$, else $z_n=x_n+y_n+carry-b$ and $carry=1$.
To formalize the above algorithm, introduce the carry as a new sequence, $c_n$, and use the Euclidean division: for each n, there is a unique $c_n$ and $z_n$, such that $c_n \cdot b + z_n = x_{n+1} + y_{n+1} + c_{n+1}$.
This defines $z_n$ and $c_n$ unambiguously. However, this set of formula goes backwards (giving you $z_n$ and $c_n$ as a function of of $x_{n+1}$, $y_{n+1}$ and not the reverse). So, to prove unambiguity, you cannot use recurrence, but have to use infinite descent.
They can be used as recurrence algorithms, but with a complication due to the fact that at each step n you may have to revise the value of the last digit of the current approximation, changing it from $z_{n-1}$ to $z_{n-1}+1$ when $x_n+y_n>b$.
In order to do this neatly, look to $z_n$ and $c_n$ not as sequence, but as sequences of sequences $z_k = (z_{k,n})_{n < k}$ and $c_k = (c_{k,n})_{n < k}$, where $z_{k,n}=z_n$ and $z_{k,n}=z_n$ for each $n < k$. Then, you can use recurrence on $k$ and compute a unique $z_{k+1}$, $c_{k+1}$ staring from an arbitrary end carry $c_{k,k+1}=0, 1$, which has no impact on lower indexes.