[JEE Advanced 1978] With $(m,n)=\dfrac{(1-x^m)(1-x^{m-1})\cdots(1-x^{m-n+1})}{(1-x)(1-x^2)\cdots(1-x^n)}$ ($n\le m$), show $(m,n+1)=(m-1,n+1)+x^{m-n-1}(m-1,n)$.
With $(m,n)=\dfrac{(1-x^m)(1-x^{m-1})\cdots(1-x^{m-n+1})}{(1-x)(1-x^2)\cdots(1-x^n)}$ ($n\le m$), show $(m,n+1)=(m-1,n+1)+x^{m-n-1}(m-1,n)$.
1 Answer
VAVidaara Admin
✓ Vidaara Team
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· 1mo ago
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Answer: Proved (Gaussian binomial recurrence).
$(m,n)$ is the Gaussian binomial coefficient $\binom mn_x$; the stated identity is its standard Pascal-type recurrence, verified by expanding both sides.
JEE Advanced 1978 · Quadratic Equations and Inequations — verified solution by the Vidaara Team.
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