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Let F be a finite extension of ℚp. We show that every Schwartz function on F, with values in ℚp, is the p-adic uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof uses the notion of q-binomial coefficients and some classical identities between these polynomials. [ABSTRACT FROM AUTHOR]

We introduce new q-tangent numbers based on the Carlitz q-analog of the Eulerian polynomial and the so-called doubloon combinatorial set-up. Those new q-tangent numbers are polynomials with positive integral coefficients. They are divisible by products of binomials of the form 1 + qi, the quotients being q-analogs of the reduced tangent numbers having an explicit combinatorial interpretation. [ABSTRACT FROM PUBLISHER]

In this paper, we consider translation and multiplication operators acting on the rings of symmetric and nonsymmetric polynomials and study their matrix coefficients with respect to the bases of Jack polynomials and interpolation polynomials. The main new insight is that the symmetric and nonsymmetric cases share a key combinatorial feature, that of a locally finite graded poset with a minimum element. This allows us to treat both cases in a simple and unified manner. [ABSTRACT FROM PUBLISHER]

For a prime p and binomial axk + bxl with 1 ≤ l < k < 1/32(p – 1)2/3, we use Stepanov's method to obtain the bound |Σp−1x=1ep(axk + bxl)| ≪ max {1, lΔ−1/3}1/4 k1/4 p3/4, where Δ = (k – l)/(k, l, p – 1). [ABSTRACT FROM PUBLISHER]

Let σ(m) be the sum of divisors of the positive integer m. Here, we show that 1N+1∑0≤k≤Nσ((Nk))(Nk)≫log log logN, and that 12N∑k=0 NσNk→∞, as N tends to infinity. [ABSTRACT FROM PUBLISHER]