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Proof of two conjectures of Guo and Schlosser

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Abstract

Let \([n]=(1-q^n)/(1-q)\) denote the q-integer and \(\Phi _n(q)\) the nth cyclotomic polynomial in q. Recently, Guo and Schlosser provided two conjectures: For any odd integer \(n>3\), modulo \([n]\Phi _n(q)(1-aq^n)(a-q^n)\),

$$\begin{aligned} \sum _{k=0}^{(n+1)/2}[4k+1]\frac{(aq^{-1};q^2)_k(q^{-1}/a;q^2)_k(q;q^2)_k^2}{(aq^4;q^2)_k(q^4/a;q^2)_k(q^2;q^2)_k^2}q^{4k} \equiv 0, \end{aligned}$$

and modulo \(\Phi _n(q)^2(1-aq^n)(a-q^n)\),

$$\begin{aligned} \sum _{k=0}^{(n+1)/2}[4k+1]\frac{(aq^{-1};q^2)_k(q^{-1}/a;q^2)_k(q^{-1};q^2)_k(q;q^2)_k}{(aq^4;q^2)_k(q^4/a;q^2)_k(q^4;q^2)_k(q^2;q^2)_k}q^{6k} \equiv 0, \end{aligned}$$

where \((a;q)_k=(1-a)(1-aq)\cdots (1-aq^{k-1})\). In this paper, we confirm these two conjectures and further give their generalizations involving two free parameters. Our proof uses Guo and Zudilin’s ‘creative microscoping’ method and the Chinese remainder theorem for coprime polynomials.

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Acknowledgements

The authors thanks the anonymous referee for many valuable comments on a previous version of this manuscript.

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Menglin Yu & Xiaoxia Wang

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  1. Menglin Yu
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  2. Xiaoxia Wang
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Correspondence to Xiaoxia Wang.

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This work is supported by National Natural Science Foundations of China (11661032)

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Yu, M., Wang, X. Proof of two conjectures of Guo and Schlosser. Ramanujan J 58, 239–252 (2022). https://doi.org/10.1007/s11139-021-00452-5

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