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QJEEAdv78Mad5d6: JEE Advanced 1978 Mathematics - Class 11 JEEAdv78Mad5d6 Solution: If where and are positive integers show that

If (m , n)=(1−xm)(1−xm−1).......(1−xm−n+1)(1−x)(1−x2).........(1−xn)\left( {m\,,\,n} \right) = {{\left( {1 - {x^m}} \right)\left( {1 - {x^{m - 1}}} \right).......\left( {1 - {x^{m - n + 1}}} \right)} \over {\left( {1 - x} \right)\left( {1 - {x^2}} \right).........\left( {1 - {x^n}} \right)}}

where mm and nn are positive integers (n≤m),\left( {n \le m} \right), show that
(m,n+1)=(m−1, n+1)+xm−n−1(m−1,n).\left( {m,n + 1} \right) = \left( {m - 1,\,n + 1} \right) + {x^{m - n - 1}}\left( {m - 1,n} \right).

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