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Register a new userAn inductive proof of Straubs q-analogue of Ljunggrens congruence
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Bo Ning
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Recently, Straub gave an interesting $q$-analogue of a binomial congruence of Ljunggren. In this note we give an inductive proof of his result.
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In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}equiv (2q+2q^{-1}-1)[n]_{q^2}^4pmod{[n]_{q^2}^4Phi_n(q^2)}, $$ where $[n]=[n]_q=(1-q^n)/(1-q),(a;q)_0=1,(a;q)_k=(1-a)(1-aq)cdots(1-aq^{k-1})$ for $kgeq 1$ and $Phi_n(q)$ denotes the $n$-th cyclotomic polynomial.
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