A q-analog of Ljunggren's binomial congruence

We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $$\binom{a p}{b p} \equiv \binom{a}{b}$$ modulo $p^3$ for primes $p\ge5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark.Comment: 6 pages, to be published in the proceedings of FPSAC 201

Multivariate Ap\'ery numbers and supercongruences of rational functions

One of the many remarkable properties of the Ap\'ery numbers $A (n)$, introduced in Ap\'ery's proof of the irrationality of $\zeta (3)$, is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes $p \geq 5$. Similar congruences are conjectured to hold for all Ap\'ery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Ap\'ery numbers by showing that they extend to all Taylor coefficients $A (n_1, n_2, n_3, n_4)$ of the rational function \begin{equation*} \frac{1}{(1 - x_1 - x_2) (1 - x_3 - x_4) - x_1 x_2 x_3 x_4} . \end{equation*} The Ap\'ery numbers are the diagonal coefficients of this function, which is simpler than previously known rational functions with this property. Our main result offers analogous results for an infinite family of sequences, indexed by partitions $\lambda$, which also includes the Franel and Yang--Zudilin numbers as well as the Ap\'ery numbers corresponding to $\zeta (2)$. Using the example of the Almkvist--Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Ap\'ery-like sequences.Comment: 19 page

Anatomy of flavour-changing Z couplings in models with partial compositeness

In models with partially composite quarks, the couplings of quarks to the Z boson generically receive non-universal corrections that are not only constrained by electroweak precision tests but also lead to flavour-changing neutral currents at tree level. The impact of these flavour-changing couplings on rare K and B decays is studied in two-site models for three scenarios: an anarchic strong sector with two different choices of fermion representations both leading to a custodial protection of the Z->bb coupling, and for a strong sector invariant under a U(2)^3 flavour symmetry. In the complete numerical analysis, all relevant constraints from Delta(F)=2 processes are taken into account. In all scenarios, visible effects in rare K and B decays like K->pi nu anti-nu, B(s)->mu+mu- and B->K*mu+mu- are possible that can be scrutinized experimentally in the near future. Characteristic correlations between observables allow to distinguish the different cases. To sample the large parameter space of the anarchic models, a new method is presented that allows larger statistics than conventional approaches.Comment: 22 pages, 4 figures. v2: minor clarifications, conclusions unchanged. Matches journal versio

Informal Sector: The Credit Market Channel

We build a model of firms’ choice between formality and informality. Complying with costly registration procedures allows the firms to benefit from key public goods, enforcement of property rights and contracts, that make the participation in the formal credit market possible. In a moral hazard framework with credit rationing, their decision is shaped by the interaction between the cost of entry into formality, and the relative efficiency of formal versus informal credit mechanisms and their related institutional arrangements. The model is consistent with existing stylized facts on the determinants of informality.Formal and Informal Sectors, Credit Markets, Institutional Arrangements.

Positivity of rational functions and their diagonals

The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szeg\H{o} as well as Askey and Gasper, who inspired more recent work. It is well known that the diagonal coefficients of rational functions are $D$-finite. This note is motivated by the observation that, for several of the rational functions whose positivity has received special attention, the diagonal terms in fact have arithmetic significance and arise from differential equations that have modular parametrization. In each of these cases, this allows us to conclude that the diagonal is positive. Further inspired by a result of Gillis, Reznick and Zeilberger, we investigate the relation between positivity of a rational function and the positivity of its diagonal.Comment: 16 page