A Simple Probabilistic Proof for the Alternating Convolution of the Central Binomial Coefficients

ABSTRACT

This note presents a simple probabilistic proof of the identity for the alternating convolution of the central binomial coefficients. The proof of the identity involves the computation of moments of order n for the product of standard normal random variables.

Keywords:

• Binomial identities
• Combinatorial identities
• Probabilistic proofs
• Standard normal distribution

Acknowledgments

The author is grateful to the editor, associate editor, and the referee for helpful comments and suggestions, which improved the presentation of the result. This work was initiated while the author was a post-doctoral fellow (Project Code No. 15DSTINS010) with Prof. Koushik Saha at IIT Bombay, Mumbai, India in the recent past. The author wishes to thank Prof. Koushik Saha for useful comments.