142 research outputs found
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
A -microscope for supercongruences
By examining asymptotic behavior of certain infinite basic (-)
hypergeometric sums at roots of unity (that is, at a "-microscopic" level)
we prove polynomial congruences for their truncations. The latter reduce to
non-trivial (super)congruences for truncated ordinary hypergeometric sums,
which have been observed numerically and proven rarely. A typical example
includes derivation, from a -analogue of Ramanujan's formula of the two supercongruences valid
for all primes , where denotes the truncation of the infinite sum
at the -th place and stands for the quadratic character
modulo .Comment: 26 page
Extending Gaussian hypergeometric series to the -adic setting
We define a function which extends Gaussian hypergeometric series to the
-adic setting. This new function allows results involving Gaussian
hypergeometric series to be extended to a wider class of primes. We demonstrate
this by providing various congruences between the function and truncated
classical hypergeometric series. These congruences provide a framework for
proving the supercongruence conjectures of Rodriguez-Villegas.Comment: Int. J. Number Theory, accepted for publicatio
Some q-analogues of supercongruences of Rodriguez-Villegas
We study different q-analogues and generalizations of the ex-conjectures of
Rodriguez-Villegas. For example, for any odd prime p, we show that the known
congruence \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv
(-1)^{\frac{p-1}{2}}\pmod{p^2} has the following two nice q-analogues with
[p]=1+q+...+q^{p-1}:
\sum_{k=0}^{p-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\varepsilon)k} &\equiv
(-1)^{\frac{p-1}{2}}q^{\frac{(p^2-1)\varepsilon}{4}}\pmod{[p]^2}, where
(a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and
\varepsilon=\pm1. Several related conjectures are also proposed.Comment: 14 pages, to appear in J. Number Theor
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