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

Authors

Department of Mathematics, Wenzhou University, Wenzhou, PR China

Abstract

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

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References

[1] G. E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998.
[2] A. Berkovich and S.O. Warnaar, Positivity preserving transformations for q-binomial coefficients, Trans. Amer. Math. Soc., 357 (2005) 2291–2351.
[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.

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History

  • Receive Date: 29 September 2022
  • Revise Date: 01 February 2023
  • Accept Date: 05 February 2023
  • Published Online: 01 June 2024

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