q -Supercongruences from Transformation Formulas
Results in Mathematics(2022)
Abstract
Let Φ _n(q) denote the n -th cyclotomic polynomial in q . Recently, Guo and Schlosser (Constr Approx 53:155–200, 2021) put forward the following conjecture: for any odd integer n>1 , ∑ _k=0^n-1[8k-1](q^-1;q^4)_k^6(q^2;q^2)_2k/(q^4;q^4)_k^6(q^-1;q^2)_2kq^8k ≡{[ 0 (mod [n]Φ _n(q)^2), if n≡ 1 (mod 4),; 0 (mod [n]), if n≡ 3 (mod 4). ]. where (a;q)_k=(1-a)(1-aq)… (1-aq^k-1) , [n]=(1-q^n)/(1-q) , and Φ _n(q) denotes the n -th cyclotomic polynomial in q . Applying the ‘creative microscoping’ method and several summation and transformation formulas for basic hypergeometric series and the Chinese remainder theorem for coprime polynomials, we confirm the above conjecture, as well as another similar q -supercongruence conjectured by Guo and Schlosser.
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Key words
Congruence, cyclotomic polynomial,
q-binomial coefficient, Watson’s transformation,
q-Pfaff–Saalschütz summation, creative microscoping, the Chinese remainder theorem., Primary 11B65, Secondary 05A10, 05A30, 11A07
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