A new $q$-analogue of the binomial identity $\sum_{k}(-1)^k{2n\choose n+3k}=2\cdot 3^{n-1}$

Document Type : Research Paper


Department of Mathematics, Wenzhou University, Wenzhou, PR China


In this paper, we establish a new $q$-analogue of the binomial identity:
&\sum_{k}(-1)^k{2n\choose n+3k}=
1,&\text{if $n=0$,}\\[5pt]
2\cdot3^{n-1},&\text{if $n\ge 1$.}
Our proof relies on a weight-preserving and sign-reversing involution due to Guo and Zhang.


Main Subjects

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[3] V. J. W. Guo and J. Zhang, Combinatorial proofs of a kind of binomial and q-binomial coefficient identities, Ars Combin., 113 (2014) 415–428.
[4] M. E. H. Ismail, D. Kim and D. Stanton, Lattice paths and positive trigonometric sums, Constr. Approx., 15 (1999) 69–81.
[5] L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc., 54 (1952) 147–167.
  • Receive Date: 29 September 2022
  • Revise Date: 01 February 2023
  • Accept Date: 05 February 2023
  • Published Online: 01 June 2024