Arithmetic properties of q-Fibonacci numbers and q-Pell numbers

We investigate some arithmetic properties of the q-Fibonacci numbers and the q-Pell numbers.Comment: 12 page

q-Analogues of Wilson's theorem

Electronic version of an article published in International Journal of Number Theory Volume 04, Issue 04, August 2008, pp. 539-547. DOI: 10.1142/S1793042108001511. Copyright © 2008 World Scientific Publishing Company: http://www.worldscientific.com/worldscinet/ijntWe give q-analogues of Wilson's theorem for the primes congruent to 1 and 3 modulo 4, respectively. Also q-analogues of two congruences due to Mordell and Chowla are established

q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the $q$-Lucas numbers.Comment: Incorporated comments from referees. Accepted for publication in Int. J. Number Theor