A Binomial Identity via Differential Equations

In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order. Several proofs of the well-known identity n ∑ k=0 ( n + k n ) 2 = 2 (1) [4, (1.79)] appear in the literature. In [3, Equation (5.20)], it is proved using partial sums of binomial series. In [6], (1) is verified by a probabilistic argument. Another short probabilistic proof follows from [9] by taking p = 1/2. In [2, p. 64, Ex. 2], the sum in (1) is calculated by means of contour integration in the complex plane. In the electronic manuscript [8, p. 62], the equivalent n ∑ k=0 ( 2n − k n − k ) 2 = 2 is shown by means of Riordan arrays and the Lagrange Inversion Formula. In addition, one referee of this manuscript suggested another proof by means of generating functions and the Lagrange Inversion Formula. Another referee was kind enough to point out the similarity between (1) and the identity