Gaussian Binomial Theorem/Negation of Upper Index
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Theorem
Let $r \in \R$ be a real number.
\(\ds \prod_{k \mathop \ge 0} \dfrac {1 + q^{k + r + 1} x} {1 + q^k x}\) | \(=\) | \(\ds \sum_{k \mathop \in \Z} \dbinom {-r - 1} k_q q^{k \left({k - 1}\right) / 2} \left({-q^{r + 1} x}\right)^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in \Z} \dbinom {k + r} x^k\) |
where:
- $\dbinom r k_q$ denotes a Gaussian binomial coefficient.
- $x \in \R: \left\lvert{x}\right\rvert < 1$
- $q \in \R: \left\lvert{q}\right\rvert < 1$.
Proof
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Source of Name
This entry was named for Carl Friedrich Gauss.
Sources
- 1843: Augustin Louis Cauchy: Mémoire sur les fonctions dont plusieurs valeurs sont liées entre elles par une équation linéaire, et sur diverses transformations de produits composés d'un nombre indéfini des facteurs (Comptes rendus de l'Académie des Sciences Vol. 17: pp. 523 – 531)
- 1847: E. Heine: Untersuchungen über die Reihe (J. reine angew. Math. Vol. 34: pp. 285 – 328)
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $58$