Abstract First published in 1961, Twersky’s formula provides an alternative representation for a class of Schlömilch series that is important in wave scattering theory. When Twersky’s formula is used in practical applications, its rate of convergence is often accelerated using Kummer’s transformation. This amounts to expanding the summand in negative powers of the index, subtracting these terms and adding back their exact sum. The first few terms can easily be obtained using a computer algebra system, but they become increasingly complicated as the expansion progresses. In this paper, we derive a general expression for the coefficients in the expansion of the summand. We also show that these coefficients can be calculated using a stable, linear recurrence relation. Using these results it is possible to create simple and very rapidly convergent representations for the Schlömilch series.

We provide an elementary proof of the fact that a sequence defined by a linear recurrence relation with integer coefficients is periodic if and only if all characteristic roots are distinct roots of unity. Additionally, we discuss the case in which the coefficients of the recurrence relation are restricted to the set {–1,0,1}.

We investigate the multifractal geometry of irregular sets arising from weighted averages of random variables, where the weights (wn) form a positive sequence with exponential growth. Our analysis applies in particular to sequences generated by linear recurrence relations of Fibonacci type, including higher-order generalizations such as the Tetranacci sequence (Tn). Using a Cantor-type construction built from alternating free and forced blocks, we show that the associated exceptional sets may attain full Hausdorff and packing dimension, independently of the precise form of the recurrence. We further develop a probabilistic interpretation of (Tn) through an appropriate Markov representation that encodes its combinatorial evolution and yields sharp asymptotic behavior. Finally, given n+1 consecutive terms of a Fibonacci-type sequence, one may construct a polynomial Pn(x) of degree at most n via Lagrange interpolation; we show that this polynomial admits an implicit recursive representation consistent with the underlying recurrence.

The Tribonacci-Lucas sequence {𝑆 𝑛 } ≥0 is defined by the linear recurrence relation 𝑆 𝑛+3 = 𝑆 𝑛+2 + 𝑆 𝑛+1 + 𝑆 𝑛 , for 𝑛 ≥ 0, with the initial conditions 𝑆 0 = 𝑆 2 = 3 and 𝑆 1 = 1. A palindromic number is a number that remains the same when its digits are reversed. This paper uses Baker’s theory for nozero lower bounds for linear forms in logarithms of algebraic numbers, and reduction methods involving the theory of continued fraction to determine all Tribonacci-Lucas numbers that are palindromic concatenations of two distinct repdigits.

We derive a Binet-type formula for operator-valued sequences satisfying linear recurrence relations, extending the classical scalar case to the setting of bounded operators on Hilbert spaces. In this framework, we analyze the operator moment problem as an application, establishing new connections between recursive operator sequences and moment sequences.

In this paper, we investigate the stability of finite difference schemes for linear partial differential equations (PDEs) in the sense of Ulam-Hyers. First, we prove a general theorem that connects the consistency, convergence and Ulam-Hyers stability of the finite difference scheme. We establish conditions under which the scheme is Ulam-Hyers stable by converting the difference equation into an equivalent linear recurrence relation with matrix coefficients. We then prove that Ulam-Hyers stability is necessary for the convergence of the scheme. Finally, we show that consistency and Ulam-Hyers stability together imply the convergence of the finite difference scheme. To illustrate the applicability of our results, we present a Ulam-Hyers stability and convergence analysis for finite difference schemes applied to linear parabolic and hyperbolic PDEs. Two numerical examples are provided to validate the theoretical findings.

This paper introduces the Generalized Tribonacci Hybrid Split Quaternion (GTHSQ), a novel split quaternion with coefficients derived from generalized Tribonacci hybrid numbers. This form unifies various existing number types, such as generalized hybrid Tribonacci numbers and Tribonacci numbers, offering a fresh perspective on quaternion theory. To systematically characterize GTHSQ, we develop a comprehensive mathematical framework. This includes defining GTHSQ through related number expressions, specifying its operational rules, and proving that it retains the third-order linear recurrence relation of generalized Tribonacci sequences. We derive its Binet-type formula and generating functions while exploring its core properties. Additionally, we extend classic combinatorial identities (Vajda, Catalan, Cassini) to GTHSQ, define the GTHSQ matrix, and analyze its product with the S-matrix—a generalized third-order linear recurrence sequence representation matrix—to obtain matrix and determinant expressions for GTHSQ. These findings verify the closed-form solution of GTHSQ in terms of combinatorial identities and matrix representation. Furthermore, we discuss potential applications of GTHSQ, including advancements in quaternion algebra, support for encryption algorithms in cryptography, and simplification of spatial transformations in physics, thereby providing new tools and insights for future research in quaternion theory and interdisciplinary studies.

Generalised power series determined by linear recurrence relations

Abstract Quantum secret sharing plays an essential role in quantum cryptography. This paper proposes a dynamic hierarchical quantum multi-secret sharing scheme based on linear homogeneous recurrence (LHR) relations. In the scheme, the distributor aggregates multiple secrets into a single master secret using the Chinese Remainder theorem (CRT), and then implements a special hierarchical structure based on LHR relations, generating corresponding shares for the participants. Participants use the generalized Pauli operators to encode their shares into the quantum state to reconstruct multiple secrets. During the transmission of the quantum state, participants can authenticate each other, thereby enhancing the security of the communication process. Additionally, the proposed scheme is dynamic, allowing for the dynamic update of the participants’ set. Security analysis demonstrates that the scheme can resist measurement attacks, intercept-resend attacks, entanglement-measurement attacks, Trojan horse attacks, forgery attacks, collusion attacks, as well as revoked or downgraded participant attacks.

The beautiful Beraha–Kahane–Weiss (BKW) theorem has found many applications within graph theory, allowing for the determination of the limits of zeros of graph polynomials in a wide range of settings such as chromatic polynomials, network reliability, and generating polynomials related to independence and domination. However, the proof only provides solutions for linear recurrence relations of polynomials whose characteristic polynomials have simple zeros. Here we extend the class of functions to which the BKW theorem can be applied, and provide some applications in combinatorics.

The sequence known as Narayana’s cows sequence is defined by the thirdorder linear recurrence relation Nn = Nn−1 + Nn−3, for n ≥ 3, with the initial values given by N0 = N1 = N2 = 1. In this paper, we explore the class of numbers known as brepdigits that can be represented as the sum of three terms from this sequence. Specifically, we aim to identify all such numbers for bases in the range 2 ≤ b ≤ 30. Our approach relies on precise lower bounds for linear forms in logarithms, along with an enhanced version of the Baker-Davenport reduction method applied to Diophantine equations.

This paper explores certain properties of the par and par^{+} functions of Toeplitz matrices. These functions are studied in tandem due to their many shared properties. Since the combinatorial foundation of these functions lies in ordered partitions of a natural number into non-negative integers, it becomes possible to represent them as partition polynomials and to construct recursive algorithms for their computation. In addition to a brief introduction to these functions, the paper presents a recurrence relation for computing the par-functions of Toeplitz matrices, which enables the unification of a broad class of linear recurrence relations. As linear recurrence relations are often related to partition polynomials, the representation of these functions as partition polynomials is also studied. The article includes an example that utilizes the fact that the multilinear polynomials of par^+ and par-functions of square matrices contain 2^(n-1) terms, with the par-function comprising half positive and half negative terms. Two combinatorial identities are derived using a Toeplitz matrix whose entries are all equal to one.

In this study, we define a new family of Gaussian polynomials, called Gaussian Chebyshev polynomials, by extending classical Chebyshev polynomials into the complex domain. These polynomials are characterized by second-order linear recurrence relations, and their connections with the Chebyshev polynomials are established. We also examine properties such as Binet-type formulas and generating functions. Moreover, we characterize some relationships between Gaussian and classical Chebyshev polynomials for the first and second kinds. We obtain some well-known theorems, such as Cassini, Catalan, and d’Ocagne’s theorems, for the first and second kinds. Furthermore, we present important connections among four types of these new polynomials. In the proofs of our results, we utilize the symmetric and antisymmetric properties of the Chebyshev polynomials. Finally, it is shown that Gaussian Chebyshev polynomials are closely related to well-known special sequences such as the Fibonacci, Lucas, Gaussian Fibonacci, and Gaussian Lucas numbers for some specific values of variables.

In this paper, generalization ${{V}_{n}}\left( {{V}_{i}},{{p}_{i}} \right)=\sum\limits_{j=1}^{3}{{{p}_{1}}{{V}_{n-j}}},\text{ }j=1,2,3,\text{ }n\ge 4$ of the third order linear recurrence relations is considered and represented in two different ways to generate Tribonacci numbers. Thereby, a matrix representation is established to engender numbers and to investigate identities and results in the generalized form. It is observed from the obtained generalized results that relevant previous outcomes become the special cases. On picking up ${{V}_{j},{p}_{j}},\text{ }j=1,2,3$ initial terms, coefficients of the linear recurrence relations and the power $n$ of the matrix${{M}^{n}}$ arbitrarily, to demonstrate the communication of message. For secruty, same message can be send frequently with a different paprmetres by varying ${{V}_{j},{p}_{j}},\text{ }j=1,2,3$ and $n$ . A numerical example is presented to illustrate to forward and retrieve the messages.

The generalized Petersen graph <span class="math inline">\(G(n,k)\)</span> is a cubic graph with vertex set <span class="math inline">\(V(G(n,k))=\{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}\)</span> and edge set <span class="math inline">\(E(G(n,k))=\{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}\)</span> where the indices are taken modulo <span class="math inline">\(n\)</span>. Schwenk found the number of Hamiltonian cycles in <span class="math inline">\(G(n,2)\)</span>, and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in <span class="math inline">\(G(n,3)\)</span> and <span class="math inline">\(G(n,4)\)</span>. This is attained by introducing <span class="math inline">\(G'(n,k)\)</span>, which is a modified version of <span class="math inline">\(G(n,k)\)</span>, and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for <span class="math inline">\(k=3\)</span> and <span class="math inline">\(k=4\)</span>, which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in <span class="math inline">\(G(n,k)\)</span> is a sum of the number of admissible subgraphs of <span class="math inline">\(G'(n,k)\)</span> over a certain subset of the classes.

In this article, we study the sequence {Vn}, which is generated by the (p, q)-Generalised linear recurrence relation of second order Vn(p, q, a, b) = pVn-1 + qVn-2, n ≥ 2, with the initial terms V0 = a, V1 = b, where p, q, a and b (ab ≠ 0, pq ≠ 0) are arbitrary real numbers. Addition, subtraction formulas, Binet formula and some new results are obtained and studied in the generalised form. Some existing and new identities are also explored, employing this generalized definition of the sequence {Vn} and becoming the special cases, on substituting the coefficients p, q of the recurrence relation and the initial terms V0, V1.

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By establishing a connection with generalized Balancing sequences, we derive explicit formulas that describe the system’s asymptotic behavior. Our main contribution is proving the existence of a unique globally asymptotically stable equilibrium point for all trajectories, regardless of initial conditions. We also provide analytical expressions for the solutions and support our findings with numerical examples. These results offer valuable insights into the dynamics of nonlinear rational systems and form a theoretical basis for further exploration in this area.

ABSTRACTWe propose a Essential Secret Image Sharing scheme using Linear Homogeneous Recurrence Relation and polynomials for sharing a grayscale or color secret image in the semihonest model. In our scheme, the dealer generates essential and nonessential shares of a secret image. A combiner needs shares to reconstruct the secret image, where at least are essential shares. Unlike most state‐of‐the‐art schemes restricting to be equal to , our scheme also allows for . This merit makes reconstruction possible even if up to essential shares are unavailable. Additionally, compared to state‐of‐the‐art schemes, our scheme offers substantial reductions in share sizes—by factors formed from , , , and . Thus, with this reduced size of shares, leading to reduced share storage costs, our scheme has a broader range of applications, including those with limited budgets. Moreover, in cases where , the shares generation period in our scheme, during which an adversary can potentially steal the secret image from the dealer, is at least 42% shorter than that in the state‐of‐the‐art scheme supporting .

ABSTRACTAvicenna numbers that we define in this paper, are a class of figurate numbers, including icosahedral, octahedral, tetrahedral, dodecahedral, rhombicosidodecahedral numbers and cubes, play a key role in mathematics, physics and various scientific fields. These numbers describe point arrangements in symmetrical 3D shapes and are essential in number theory, combinatorics, and geometric symmetry. They have practical applications in fields like chemistry, biology, materials science, virology, and network theory, helping model atomic arrangements, molecular structures, crystal lattices, viral capsids, and optimal networks. In this paper, we give some mathematical properties of Avicenna numbers that includes platonic and Archimedean numbers. Avicenna numbers satisfy fourth‐order linear recurrence relations. Also, we mention their applications and relations to mathematics, physics and other sciences.

Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and remarkable results concerning power sums, ordinary convolutions, and binomial convolutions are presented by employing the symmetric feature, series rearrangements, and the generating function approach. Most of the classical results concerning these six number/polynomial sequences are recorded as consequences.