[Math] Is the term “telescoping product” well known

productssequences-and-series

I know that "telescoping series" (or sum) is well known. But I can't find many reliable references to the term "telescoping product". It would be one of the following:

  • $x_i = \dfrac{y_i}{y_{i+1}}$:
    $$\require{cancel}\begin{align}
    \prod_{i=m}^n x_i &= \prod_{i=m}^n\frac{y_i}{y_{i+1}}\\
    &= \frac{y_m}{\cancel{y_{m+1}}} \cdot \frac{\cancel{y_{m+1}}}{y_{m+2}} \,\,\cdots\,\, \frac{y_{n-1}}{\cancel{y_n}} \cdot \frac{\cancel{y_n}}{y_{n+1}}\\
    &= \frac{y_m}{y_{n+1}}
    \end{align}$$

or

  • $x_i = \dfrac{y_i}{y_{i-1}}$:
    $$\begin{align}
    \prod_{i=m}^n x_i &= \prod_{i=m}^n\frac{y_i}{y_{i-1}}\\
    &= \frac{\cancel{y_m}}{y_{m-1}} \cdot \frac{y_{m+1}}{\cancel{y_m}} \,\,\cdots\,\, \frac{\cancel{y_{n-1}}}{y_{n-2}} \cdot \frac{y_n}{\cancel{y_{n-1}}}\\
    &= \frac{y_n}{y_{m-1}}
    \end{align}$$

It shows up in signal processing, for example, in a formula to compute the modified prediction errors of RLS filters non-recursively (reducing the numerical error). But I've never seen the term "telescoping product" in any bibliography. So would it be inappropriate for me to use that term without explaining what it means?

Best Answer

From a simple google search, many teaching resources use the term "Telescoping Sums and Products":

  1. www.math.cmu.edu/.../3-telescope-solns.pdf (broken now, but from a reliable CMU edu website. Google says it is titled: III. Telescoping Sums and Products)

  2. http://faculty.wwu.edu/sarkara/ph13.pdf (Title of paper is "Telescoping Sums and Products")

Regardless, I think I could figure out what you meant...

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