21 research outputs found
Some congruences involving central q-binomial coefficients
Motivated by recent works of Sun and Tauraso, we prove some variations on the
Green-Krammer identity involving central q-binomial coefficients, such as where is
the Legendre symbol and is the th cyclotomic polynomial. As
consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q
&\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose
2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for , the
first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence
modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to
appear in Adv. Appl. Mat
Some new -congruences for truncated basic hypergeometric series
We provide several new -congruences for truncated basic hypergeometric
series, mostly of arbitrary order. Our results include congruences modulo the
square or the cube of a cyclotomic polynomial, and in some instances,
parametric generalizations thereof. These are established by a variety of
techniques including polynomial argument, creative microscoping (a method
recently introduced by the first author in collaboration with Zudilin),
Andrews' multiseries generalization of the Watson transformation, and
induction. We also give a number of related conjectures including congruences
modulo the fourth power of a cyclotomic polynomial.Comment: 14 pages, more background and references adde