Proof of two conjectures of Guo and Schlosser,The Ramanujan Journal

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Proof of two conjectures of Guo and Schlosser
The Ramanujan Journal ( IF 0.7 ) Pub Date : 2021-07-02 , DOI: 10.1007/s11139-021-00452-5
Menglin Yu ₁ , Xiaoxia Wang ₁

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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.

更新日期：2021-07-04

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