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Convolution identity - OIMU 2006 Problem 6

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August 30, 2010
real analysisalgebra proposedalgebra

Problem Statement

Let x0(t)=1x_0(t)=1, xk+1(t)=(1+tk+1)xk(t)x_{k+1}(t)=(1+t^{k+1})x_k(t) for all k≥0k\geq 0; yn,0(t)=1y_{n,0}(t)=1, yn,k(t)=tn−k+1−1tk−1yn,k−1(t)y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t) for all n≥0n\geq 0, 1≤k≤n1\leq k \leq n.
Prove that ∑j=0n−1(−1)jxn−j−1(t)yn,j(t)=1−(−1)n2\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2} for all n≥1n\geq 1.
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