Sums of Powers of Binomial Coefficients via Legendre Polynomials

Abstract

\[S_{(p,x)} = \sum\limits_{k=0}^{n} {\binom{n}{k}}^p x^k\]

where \(n \geq 0\).

Then it is well-known that \(S_n(1,x), S_2(2,1), S_n(3,1)\) and \(S_n(3,1)\) can be exhibited in closed form. The formula

\[S_{2n}{(3,-1)} = (-1)^n\binom{2n}{n}\binom{3n}{n}\]

was discovered by A. C. Dixon in \(1891\). L. Carlitz [Mathematics Magazine, Vol. \(32 (1958), 47-48]\) posed the formulas

\[S_n{(3,1)}= ((x^n))(1-x^2)^nP_n(\frac{1+x}{1-x})\]

and

\[S_n{(4,1)} = ((x^n))(1-x)^{2n}\{P_n(\frac{1+x}{1-x})\}\]

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

\[S_n{(3,-1)} = ((x^n))(1_x)^{2n}\]

and

\[S_n{(5,1)} = ((x^n))(1_x)^{2n}P_n(\frac{1+x}{1-x}S_n(3,x)\]

We obtain some partial results for \(S_n(p,x)\) when \(p\) is arbitrary, and also give a new proof of Dixon’s formula.