We consider analogs of several classical diophantine equations, such as Fermat’s last theorem and Catalan’s conjecture, for certain classes of analytic functions. We give simple direct proofs avoiding use of deep theorems in complex analysis. As a byproduct of our results, we obtain new proofs for the corresponding results over polynomials.

Applying the multisection series method to the MacLaurin series expansion of arcsin-function, we transform the Apéry–like series involving the central binomial coefficients into systems of linear equations. By resolving the  linear systems (for example, by Mathematica), we establish numerous remarkable infinite series formulae for π and logarithm functions, including several recent results due to Almkvist et al. (2003) and Zheng (2008).

We introduce the notion of capacity (ability to contain water) for compositions. Initially the compositions are  defined on a finite alphabet \([k]\) and thereafter on \(\mathbb{N}\). We find a capacity generating function for all compositions, the average capacity generating function and an asymptotic expression for the average capacity as the size of the composition increases to infinity

As suggested by Currie, we apply the probabilistic method to problems regarding pattern avoidance. Using techniques from analytic combinatorics, we calculate asymptotic mean pattern occurrence and use them in  conjunction with the probabilistic method to establish new results about the Ramsey theory of unavoidable  patterns in the abelian full word case and in the nonabelian partial word case.

In this paper, we present several explicit formulas of the sums and hypersums of the powers of the first \((n + 1)\)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences as the partial sums of the Hurwitz zeta function. These transformation coefficients satisfy many properties which are analogous to known identities and expansions of the Stirling numbers of the first kind and to the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic  denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers s ≥ 3.

Let \([k] = \{1, 2, \ldots, k\}\) be an alphabet over \(k\) letters. A word \(\omega\) of length \(n\) over alphabet \([k]\) is an element of \([k]^n\) and is also called \(k\)-ary word of length \(n\). We say that \(\omega\) contains a peak, if exists \(2 \leq i \leq n-1\) such that \(\omega_{i-1} \omega_{i+1}\). We say that \(\omega\) contains a symmetric peak, if exists \(2 \leq i \leq n-1\) such that \(\omega_{i-1} = \omega_{i+1} < \omega_i\), and contains a non-symmetric peak, otherwise. In this paper, we find an explicit formula for the generating functions for the number of \(k\)-ary words of length \(n\) according to the number of symmetric peaks and non-symmetric peaks in terms of Chebyshev polynomials of the second kind. Moreover, we find the number of symmetric and non-symmetric peaks in \(k\)-ary word of length \(n\) in two ways by using generating functions techniques, and by applying probabilistic methods.

In this paper, we first give a new \( q \)-analogue of the Lah numbers. Then we show the irreducible factors of the \( q \)-Lah numbers over \( \mathbb{Z} \).

Let \( A \) and \( B \) be additive sets of \( \mathbb{Z}_{2k} \), where \( A \) has cardinality \( k \) and \( B = v \cdot C A \) with \( v \in \mathbb{Z}_{2k}^\times \). In this note, some bounds for the cardinality of \( A + B \) are obtained using four different approaches. We also prove that in a special case, the bound is not sharp and we can recover the whole group as a sumset.