Volume 30, 2024, Number 2 (Online First)

-Dyck paths differ from ordinary Dyck paths by using an up-step of length . We analyze at which level the path is after the -th up-step and before the -st up-step. In honour of Rainer Kemp who studied a related concept 40 years ago, the terms max-terms and min-terms are used. Results are obtained by an appropriate use of trivariate generating functions; practically no combinatorial arguments are used.

We consider the negative polynomial Pell’s equation , where be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions with integer coefficients.

Let be an odd prime In Part I, starting from an -cycle in a finite field , we have established how the divisors of Mersenne, Fermat and Lehmer numbers arise. The converse question is taken up in this Part with the introduction of an arithmetic function and the notion of a split-associated prime.

In this study, we examine the Leonardo sequence with dual-generalized complex () coefficients for . Firstly, we express some summation formulas related to the Fibonacci, Lucas, and Leonardo sequences. Secondly, we present some order- characteristic relations, involving d’Ocagne’s, Catalan’s, Cassini’s, and Tagiuri’s identities. The essential point of the paper is that one can reduce the calculations of the Leonardo sequence by considering . This generalization gives the dual-complex Leonardo sequence for , hyper-dual Leonardo sequence for , and dual-hyperbolic Leonardo sequence for .

The purpose of this paper is to determine the algebraic properties of finite groups via a Mersenne-like sequence. Firstly, we introduce the generalized order -Mersenne number sequences and study the periods of these sequences modulo . Then, we get some interesting structural results. Furthermore, we expand the generalized order -Mersenne number sequences to groups and we give the definition of the generalized order -Mersenne sequences, , in the -generator groups and also, investigate these sequences in the non-Abelian finite groups in detail. At last, we obtain the periods of the generalized order -Mersenne sequences in some special groups as applications of the results produced.

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of , including their convergence criteria, and applied to many standard sequences, as particular cases of a generic . The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

Let be a non-negative integer and be a positive integer. Let be the sum of digits of written in base In 1940, Bush proved that is asymptotic to In 1968, Trollope proved an explicit formula for the error term of labeled by where is a positive integer. In 1975, Delange extended Trollope’s result to an arbitrary base by another method and labeled the error term When the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope’s method to extend to with a general base and we also proved directly that and are equal for any

Using a simple transformation, we obtain much simpler forms for some series involving binomial coefficients derived by Necdet Batır. New evaluations are given and connections with Fibonacci numbers and the golden ratio are established. Finally we derive some Fibonacci and Lucas series involving the reciprocals of .

This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, ‘m-cardinality’, is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i.e., the multiplicity of the elements). Based on m-cardinality, ‘m-cardinal numbers’ are defined as a generalization of cardinal numbers in the context of multisets. Some properties of m-cardinal numbers associated with finite and infinite msets have been researched. Concrete examples of transfinite m-cardinal numbers are given, corresponding to infinite msets which are less than ℵ₀ (the cardinality of the countably infinite set). It has been established that between finite numbers and ℵ₀ there exist hierarchies of transfinite m-cardinals, corresponding to infinite msets. Furthermore, there are examples of infinite msets with negative multiplicity that have a cardinality less than zero. We prove that there is a decreasing sequence of transfinite m-cardinal numbers, corresponding to infinite msets with negative multiplicity, and in this sequence, there is not a smallest transfinite m-cardinal number.

The arithmetic derivative is a function from the natural numbers to itself that sends all prime numbers to and satisfies the Leibniz rule. The arithmetic partial derivative with respect to a prime is the -th component of the arithmetic derivative. In this paper, we generalize the arithmetic partial derivative to -adic fields (the local case) and the arithmetic derivative to number fields (the global case). We study the dynamical system of the -adic valuation of the iterations of the arithmetic partial derivatives. We also prove that for every integer , there are infinitely many elements with exactly anti-partial derivatives. In the end, we study the -adic continuity of arithmetic derivatives.

A Lucas polynomial sequence is a pair of generalized polynomial sequences that satisfy the Lucas recurrence relation. Special cases include Fibonacci polynomials, Lucas polynomials, and Balancing polynomials. We define the -type Lucas polynomial sequences and prove that their Melham’s sums have some interesting divisibility properties. Results in this paper generalize the original Melham’s conjectures.

We offer various bounds for the divisor function , in terms of , or other arithmetical functions.

The object of this article is to establish the relation between the generating function of the quadratic form {and} the generating function{s} for the quadratic forms , , and . In the process, we deduce certain interesting theta function identities.

Let . For , a lattice point is powerful if and only if is a powerful number, where is the greatest common divisor function. In this paper, we count the number of the ordered pairs , such that the lattice point is powerful. Moreover, we study -free lattice points analogues of powerful lattice points.