In this paper, we define the q-analogue of the so-called symmetric infinite matrix algorithm. We find an explicit formula for entries in the associated matrix and also for the generating function of the k-th row of this matrix for each fixed k. This helps us to derive analytic and number theoretic identities with respect to the q-harmonic numbers and q-hyperharmonic numbers of Mansour and Shattuck.

Bargraphs are lattice paths in \(\mathbb{N}_0^2\) with three allowed types of steps: up \((0,1)\), down \((0,-1)\), and horizontal \((1,0)\). They start at the origin with an up step and terminate immediately upon return to the \(x\)-axis. A wall of size \(r\) is a maximal sequence of \(r\) adjacent up steps. In this paper, we develop the generating function for the total number of walls of fixed size \(r \geq 1\). We then derive asymptotic estimates for the mean number of such walls.

We survey four instances of the Fourier analytic ‘transference principle’or ‘dense model lemma’, which allows one to approximate an unbounded function on the integers by a bounded function with similar Fourier transform. Such a result forms a component of a general method pioneered by Green to count solutions to a single linear equation in a sparse subset of integers.

Let \( E \subseteq \mathbb{F}_q^2 \) be a set in the 2-dimensional vector space over a finite field with \( q \) elements, which satisfies \(|E| > q\). There exist \( x, y \in E \) such that \(|E \cdot (y – x)| > q/2\). In particular, \( (E + E) \cdot (E – E) = \mathbb{F}_q.\)

Let \((x(n))_{n \geq 1}\) be an \(s\)-dimensional Niederreiter-Xing sequence in base \(b\). Let \(D((x(n))_{n=1}^N)\) be the discrepancy of the sequence \((x(n))_{n=1}^N\). It is known that \(ND((x(n))_{n=1}^N) = O(\ln^s N)\) as \(N \to \infty\). In this paper, we prove that this estimate is exact. Namely, there exists a constant \(K > 0\), such that
\[
\inf_{w \in [0,1]^s} \sup_{1 \leq N \leq b^m} ND((x(n) \oplus w)_{n=1}^N) \geq K \ln^s \quad \text{ for } m = 1, 2, \ldots.
\]

We also get similar results for other explicit constructions of \((t,s)\)-sequences.

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the jth derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind.

We prove a number of properties of the generalized coefficients which lead to new recurrence relations and summation identities for the k-order harmonic number sequences. Other  applications of the generating function transformations we define in the article include new series expansions for the polylogarithm function, the alternating zeta function, and the Fourier series for the periodic Bernoulli polynomials. We conclude the article with a discussion of several specific new “almost” linear recurrence relations between the integer-order harmonic numbers and the generalized transformation coefficients, which provide new applications to studying the limiting behavior of the zeta function constants, ζ(k), at integers k ≥ 2.

Generating functions for Pell and Pell-Lucas numbers are obtained. Applications are given for some results recently obtained by Mansour [Mansour12]; by using an alternative approach that considers the action of the operator \(\delta_{e_1 e_2}^k\) to the series \(\sum_{j=0}^\infty a_j (e_1 z)^j\).