Positive Sums of the Classical Orthogonal Polynomials

An expansion as a sum of squares of Jacobi polynomials Pₙ(α ,β ) (x) is used to prove that if 0 λ α + β and β - 1 / 2, then \[( * ) ∑k = 0^n {{{(λ + 1)n - k }}{{(n - k)!}}} {{(λ + 1)_k }}{{k!}}{{P_k(α ,β ) }}(x){{P_k(β ,α )(i) }} 0, - 1 x < ∞ ,\] and the only cases of equality occur when $x = - 1$ for n odd and when λ = 0, α = - β = 1 / 2. Additional conditions are given under which $( * )$ holds and some special uses, limit cases, and important applications are pointed out. In particular, the case λ = α + β of $( * )$ is used to prove that if α, β - 1 / 2 then the Cesàro (C,α + β + 2) means of the Jacobi series of a nonnegative function are nonnegative. Also, it is shown that \[{d}{{dθ }}∑k = 0^n {{{(λ + 1)n - k }}{{(n - k)!}}} {{(λ + 1)_k }}{{k!}}{{sin (k + 1)θ }}{{(k + 1)sin ({θ / 2})}} < 0, 1 < θ < π , 0 λ 1,\] which extends a recent result of Askey and Steinig.