Sums of Powers of Binomial Coefficients via Legendre Polynomials

Abstract

$$S_{(p,x)}=\sum _{k=0}^n{(\genfrac{}{}{0ex}{}{n}{k})}^p{x}^k$$

where $n\ge 0$.

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

$$S_{2n}(3,-1)=(-1{)}^n(\genfrac{}{}{0ex}{}{2n}{n})(\genfrac{}{}{0ex}{}{3n}{n})$$

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

$$S_n(3,1)=((x^n))(1-x^2{)}^n{P}_n(\frac{1+x}{1-x})$$

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.