Simpson's Dissection
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Theorem
Let $\omega = e^{2 i \pi / q}$ be a primitive $q$th root of unity.
Let $p \not \equiv 0 \pmod q$.
Let:
- $\ds \map f x = \sum_{n \mathop = 0}^\infty a_n x^n$
Then:
- $\ds \sum_{n \mathop = 0}^\infty a_{n q + p} x^{n q + p} = \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^{- j p} \map f {\omega^j x}$
Proof
Expanding the sum on the right hand side, we obtain:
\(\ds \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^{- j p} \map f {\omega^j x}\) | \(=\) | \(\ds \frac 1 q \times \omega^0 \times \paren {a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 q \times \omega^{-p} \times \paren {a_0 + a_1 x \omega + a_2 x^2 \omega^2 + a_3 x^3 \omega^3 + \cdots}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 q \times \omega^{-2 p} \times \paren {a_0 + a_1 x \paren {\omega^2} + a_2 x^2 \paren {\omega^2}^2 + a_3 x^3 \paren {\omega^2}^3 + \cdots}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \cdots\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac 1 q \times \omega^{-\paren {q - 1} p} \times \paren {a_0 + a_1 x \paren {\omega^{q - 1} } + a_2 x^2 \paren {\omega^{q - 1} }^2 + a_3 x^3 \paren {\omega^{q - 1} }^3 + \cdots}\) |
Summing down the $q$ values of the term with exponent n in the array shown above, we obtain:
- $\ds \dfrac 1 q a_n x^n \times \paren {1 + \omega^{1 \paren {n - p} } + \omega^{2 \paren {n - p} } + \omega^{3 \paren {n - p} } + \cdots + \omega^{\paren {q - 1} \paren {n - p} } }$
From Sum of Powers of Primitive Complex Roots of Unity, we have:
\(\ds \sum_{j \mathop = 0}^{q - 1} \omega^{j \paren {n - p} }\) | \(=\) | \(\ds 1 + \omega^\paren {n - p} + \omega^{2 \paren {n - p} } + \cdots + \omega^{\paren {q - 1} \paren {n - p} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} q & : q \divides \paren {n - p} \\ 0 & : q \nmid \paren {n - p} \end {cases}\) |
When $n \equiv p \pmod q$:
- $\ds \dfrac 1 q a_n x^n \times \paren {1 + \omega^{1 \paren {n - p} } + \omega^{2 \paren {n - p} } + \omega^{3 \paren {n - p} } + \cdots + \omega^{\paren {q - 1} \paren {n - p} } } = a_n x^n$
When $n \not \equiv p \pmod q$:
- $\ds \dfrac 1 q a_n x^n \times \paren {1 + \omega^{1 \paren {n - p} } + \omega^{2 \paren {n - p} } + \omega^{3 \paren {n - p} } + \cdots + \omega^{\paren {q - 1} \paren {n - p} } } = 0$
Hence:
\(\ds \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^{- j p} \map f {\omega^j x}\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty a_{n q + p} x^{n q + p}\) |
$\blacksquare$
Source of Name
This entry was named for Thomas Simpson.
Sources
- 1757: Thomas Simpson: The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, &c. term of a series, taken in order; the sum of the whole series being known (Phil. Trans. Vol. 50: pp. 757 – 769)
- 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions: Chapter $\text {1}$. The Gamma and Beta Functions
- This article incorporates material from extracting every nth term of a series on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.