Sums of powers of binomial coefficients via Legendre polynomials

Define S-n(p,x) = Sigma(n) ((n)(k))(p) x(k), where n greater than or equal to 0, (k=0) Then it is well-known that S-n (1,x), S-n (2,1), S-n (2,-1), and S-n (3,-1) can be exhibited in closed form. The formula S-2n (3,-1) = (-1)(n) ((2n)(n)) ((3n)(n)) was discovered by A. C. Dixon in 1891. L. Carlitz [Mathematics Magazine, Vol. 32(1958), 47-48] posed the formulas S-n (3,1) = ((x(n))) (1-x(2))(n) P-n((1+x)(1-x)) and S-n (4,1) = ((X-n)) (1-x)(2n) {P-n((1+X)(1-x))}(2), where ((x(n))) f(x) means the coefficient of x(n) in the series expansion of f(x). We use Legendre polynomials to get the analogous formulas S-n(3,-1) = ((x(n))) (1-x)(2n) P-n((1+x)(1-x)), and S-n (5,1) = ((x(n))) (1-x)(n) P-n ((1+x)(1-x)) S-n (3, x). We obtain some partial results for S-n (p,x) when p is arbitrary, and also give a new proof of Dixon's formula.