---
tag: MDM8F48X18
kind: concept
name: Padé table
msc: [41A21, "30B70", "30E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Pad%C3%A9_table
same_as:
  wikidata: Q7124033
  wikipedia: Padé table
---

## Summary

The Padé table of a power series is the array of all its [[Padé approximants|1F16G4V64X]] $[m/n]$.
It is indexed by the numerator degree $m$ and the denominator degree $n$.
Its first column holds the Taylor polynomials, and its diagonal holds the diagonal approximants $[n/n]$.
Equal entries occur in square blocks, and the table is called normal when every entry occurs only once.
The table organizes convergence questions and links Padé approximants with [[continued fractions|04P2RJ4WPS]].

## Explanations

### L1 human

For a given function there is one best matching quotient of polynomials for each choice of the two degrees.
Writing these quotients in a grid, by degree of the top and degree of the bottom, gives the Padé table.
Moving down a column adds terms to the top; moving along a row allows more poles.
For $1/(1-z)$ every entry with a denominator of degree at least $1$ is $1/(1-z)$ itself, a single huge block.