A determinantal expression and a recursive relation of the Delannoy numbers

In the paper, by a general and fundamental, but non-extensively circulated, formula for derivatives of a ratio of two differentiable functions and by a recursive relation of the Hessenberg determinant, the author finds a new determinantal expression and a new recursive relation of the Delannoy numbers. Consequently, the author derives a recursive relation for computing central Delannoy numbers in terms of related Delannoy numbers.

A new method for solving the linear recurrence relation with nonhomogeneous constant coefficients in some cases

Recursive relation mainly describes the unique law satisfied by a sequence, so it plays an important role in almost all branches of mathematics. It is also one of the main algorithms commonly used in computer programming. This paper first introduces the concept of recursive relation and two common basic forms, then starts with the solution of linear recursive relation with non-homogeneous constant coefficients, gives a new solution idea, and gives a general proof. Finally, through an example, the general method and the new method given in this paper are compared and verified.

Picturing the Meaning of Scandinavian Rock Art – Graphic Representations, Archaeological Interpretations and Material Alterations

The paper presents a reflective overview of the recursive relation between the archaeological practice of picturing Scandinavian rock art in printed works since the mid-19th century, and how archaeologists have constructed its meaning. There seem to be an intimate connection between graphic representations of rock art and an interpretative bias towards the mimetic qualities of images. When picturing rock art, the identification of motifs is prioritized at the expense of the materiality of rock art. Ultimately, the production of graphic representations has influenced the antiquarian alteration of the archaeological remains. Today, major Scandinavian rock art sites are frequently painted red, with the purpose of highlighting the engraved imagery for visitor legibility. This practice transforms the materiality of stone into a visual language of graphic representations.

Infinite Nested Radicals - A Way to Express All Quantities Rational, Irrational Transcendental By a Single Integer Two

This paper proves that all mathematical quantities including fractions, roots or roots of root, transcendental quantities can be expressed by continued nested radicals using one and only one integer 2. A radical is denoted by a square root sign and nested radicals are progressive roots of radicals. Number of terms in the nested radicals can be finite or infinite. Real mathematical quantity or its reciprocal is first written as cosine of an angle which is expanded using cosine angle doubling identity into nested radicals finite or infinite depending upon the magnitude of quantity. The finite nested radicals has a fixed sequence of positive and negative terms whereas infinite nested radicals also has a sequenceof positive and negative terms but the sequence continues infinitely. How a single integer 2 can express all real quantities, depends upon its recursive relation which is unique for a quantity. Admittedly, there are innumerable mathematical quantities and in the same way, there are innumerable recursive relations distinguished by combination of positive and negative signs under the radicals. This representation of mathematical quantities is not same as representation by binary system where integer two has powers 0, 1, 2, 3…so on but in nested radicals, powers are roots of roots.

Some Diophantine Problems Related to k-Fibonacci Numbers

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

Orthorecursive expansion of unity

We study the properties of a sequence [Formula: see text] defined by the recursive relation [Formula: see text] for [Formula: see text] and [Formula: see text]. This sequence also has an alternative definition in terms of certain norm minimization in the space [Formula: see text]. We prove estimates on the growth order of [Formula: see text] and the sequence of its partial sums, infinite series identities, connecting [Formula: see text] with the harmonic numbers [Formula: see text] and also formulate some conjectures based on numerical computations.

Lower-level employees’ participation in strategy making over time.

There is a new trend towards the inclusion of lower-level employees in strategy making. Yet, such participation is challenging as lower-level employees typically lack the resources for participation, such as requisite strategy knowledge. Drawing on a practice-theoretical perspective on resources and data from a longitudinal, ethnographic case study of a participative strategy process in a large insurance company, we examine the dynamics of participation of lower-level employees. In particular, we identify a recursive relation between participation and the enactment of resources. With this finding we contribute to the understanding of participation over time, extending existing research on the resources of participation.

Some Identities for a Sequence of Unnamed Polynomials Connected with the Bell Polynomials

In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.