(q -binomial theorem) Prove that (x+y)(x+q y)(x+q^2 y) ⋯(x+q^n-1 y)=∑k=0^n( n k )q x^n-

step 1 So for the proof here via induction, we have for our base case, if n is equal to 1, well, then w

(q -binomial theorem) Prove that (x+y)(x+q y)(x+q^2 y)...

$(q$ -binomial theorem) Prove that $$(x+y)(x+q y)\left(x+q^{2} y\right) \cdots\left(x+q^{n-1} y\right)=\sum_{k=0}^{n}\left(\begin{array}{l}n \\k\end{array}\right)_{q} x^{n-k} y^{k},$$ where $$n !_{q}=\frac{\prod_{j=1}^{n}\left(1-q^{j}\right)}{(1-q)^{n}}$$ is the $q$ -factorial (cf. Theorem $7.2 .1$ replacing $q$ in $(7.14)$ with $x$ ) and$$\left(\begin{array}{l}n \\k\end{array}\right)_{q}=\frac{n !_{q}}{k !_{q}(n-k) !_{q}}$$ is the $q$ -binomial coefficient.

$(q$ -binomial theorem) Prove that
$$(x+y)(x+q y)\left(x+q^{2} y\right) \cdots\left(x+q^{n-1} y\right)=\sum_{k=0}^{n}\left(\begin{array}{l}n \\k\end{array}\right)_{q} x^{n-k} y^{k},$$
where
$$n !_{q}=\frac{\prod_{j=1}^{n}\left(1-q^{j}\right)}{(1-q)^{n}}$$
is the $q$ -factorial (cf. Theorem $7.2 .1$ replacing $q$ in $(7.14)$ with $x$ ) and$$\left(\begin{array}{l}n \\k\end{array}\right)_{q}=\frac{n !_{q}}{k !_{q}(n-k) !_{q}}$$
is the $q$ -binomial coefficient.

Key Concepts

q-Binomial Theorem

The q-binomial theorem is an extension of the classical binomial theorem where each term in the product, as well as the resulting coefficients, are modified by the parameter q. It expresses the expansion of a product of linear factors with q-scaled variables as a sum involving q-binomial coefficients. This theorem plays a crucial role in q-series and combinatorial identities, generalizing many results from ordinary algebra to the q-analogue.

q-Factorial

The q-factorial is a generalization of the usual factorial function where the product is taken over q-deformed terms. Instead of simply multiplying consecutive integers, the q-factorial incorporates factors that depend on the parameter q, reflecting a deformation of the counting process. It is defined using products of terms like (1 - q^j) and is essential in defining q-analogues of combinatorial quantities, including the q-binomial coefficients.

q-Binomial Coefficient

The q-binomial coefficient extends the concept of binomial coefficients by using q-factorials in its definition. This coefficient appears in the expansion of expressions in the q-binomial theorem, and it encapsulates combinatorial information that varies with the parameter q. Such coefficients are used to count objects in q-deformed combinatorial settings and reduce to classical binomial coefficients when q approaches 1.

Product Expansion in q-Series

Product expansion in the context of q-series involves expressing a product of terms that incorporate q-scaled variables as a sum of terms with q-binomial coefficients. This approach creates a bridge between factorization of polynomials and combinatorial identities, demonstrating how algebraic products are reflected in series expansions where the q-parameter introduces additional structure and symmetry.

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