Legendre Polynomials and an Inequality for a Combinatorial Sum

Let Pn be the Legendre polynomial of degree n. We use an estimate for the ultraspherical polynomials and a gamma function inequality to prove that |∫−1xPn(t)dt|<2π1n3/2(n≥1;−1≤x≤1) and we apply this result to obtain the combinatorial inequality ∑k=0nnk(n+k−1)/2nxk+1k+1−1n+1n/2n<2π12nn3/2(n≥1;−1≤x≤1). The factor 2/π given in both inequalities is the best possible.