On Zudilin's q-question about Schmidt's problem

We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740] derived from q-Pfaff-Saalschutz identity.Comment: 5 page

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

A note on two identities arising from enumeration of convex polyominoes

Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of ${}_3F_2$ of Gessel and Stanton.Comment: 10 pages, to appear in J. Comput. Appl. Math; minor grammatical change

Combinatorial Interpretations of the q-Faulhaber and q-Salie Coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salie's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations.Comment: 15 page

Microstructure, magneto-transport and magnetic properties of Gd-doped magnetron-sputtered amorphous carbon

The magnetic rare earth element gadolinium (Gd) was doped into thin films of amorphous carbon (hydrogenated \textit{a}-C:H, or hydrogen-free \textit{a}-C) using magnetron co-sputtering. The Gd acted as a magnetic as well as an electrical dopant, resulting in an enormous negative magnetoresistance below a temperature ($T'$). Hydrogen was introduced to control the amorphous carbon bonding structure. High-resolution electron microscopy, ion-beam analysis and Raman spectroscopy were used to characterize the influence of Gd doping on the \textit{a-}Gd$_x$C$_{1-x}$(:H$_y$) film morphology, composition, density and bonding. The films were largely amorphous and homogeneous up to $x$=22.0 at.%. As the Gd doping increased, the $sp^{2}$-bonded carbon atoms evolved from carbon chains to 6-member graphitic rings. Incorporation of H opened up the graphitic rings and stabilized a $sp^{2}$-rich carbon-chain random network. The transport properties not only depended on Gd doping, but were also very sensitive to the $sp^{2}$ ordering. Magnetic properties, such as the spin-glass freezing temperature and susceptibility, scaled with the Gd concentration.Comment: 9 figure