Typsetting a variant of division

Question

In Oliver Salazar Celis' "Adaptive Thiele interpolation," ACM Communications of Computer Algebra 56(3), Issue 221, September 2022, page 125 (Eq. 1) we find:

How does one typeset the vertical plus horizontal notation (note that the straight segments touch)?

And what does this notation mean?

Answer 1

A simple array could do the trick, but I have no idea about what it means. First time I see something like that.

Update: added the array package for better rules connections, as suggested in the comments.

\documentclass{article}
\usepackage{array} % better connections between the rules (see egreg's comment)
\usepackage[textwidth=15cm]{geometry}

\NewDocumentCommand{\myfrac}{mm}{\begin{array}{c|}#1\\\hline\multicolumn{1}{|c}{#2}\end{array}}

\begin{document}
\[
C_n(x) = \varphi_0[x_0] + \myfrac{x-x_0}{\varphi_1[x_0,x_1]} +
                          \myfrac{x-x_1}{\varphi_2[x_0,x_1,x_2]} +
                          \myfrac{x-x_2}{\cdots} +
                          \myfrac{x-x_{n-1}}{\varphi_n[x_0,\ldots,x_n]},
\]
\end{document}

Answer 2

Using holtpolt package.

\documentclass[a4paper,12pt]{article}
\usepackage{holtpolt}
\usepackage{amssymb}

\begin{document}
\[C_n(x) = \varphi_0[x_0] +\polter{x-x_0}{\varphi_1[x_0,x_1]}+\polter{x-x_1}{\varphi_2[x_0,x_1,x_2]}+\polter{x-x_2}{\ldots}+\polter{x-x_{n-1}}{\varphi_n[x_0,\ldots,x_n]}\]
\end{document}