Search

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices. [ABSTRACT FROM AUTHOR]

Proudfoot, Xu, and Young introduced the Z-polynomial for any matroid and conjectured the polynomial has only real roots. Recently, Ferroni, Nasr, and Vecchi introduced the γ-polynomial of a matroid. In this paper, we prove that both the Z-polynomials and γ-polynomials of uniform matroids are ultra log-concave, which due to Newton's inequality partially supports the real-rootedness conjecture. We also give an alternative formula for the γ-polynomials of uniform matroids. As an application, we use this formula to provide a new proof of the γ-positivity of sparse paving matroids. [ABSTRACT FROM AUTHOR]

This work aims to provide a new Galerkin algorithm for solving the fractional RayleighStokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples. [ABSTRACT FROM AUTHOR]

We consider a generalization of the classical coupon collector problem, where the set of available coupons consists of standard coupons (which can be part of the collection), and two coupons with special purposes: one that speeds up the collection process and one that slows it down. We obtain several asymptotic results related to the expectation and the variance of the waiting time until a portion of the collection is sampled, as the number of standard coupons tends to infinity. [ABSTRACT FROM AUTHOR]

This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics. [ABSTRACT FROM AUTHOR]

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments M 2 p r . The latter are expressed in terms of the 3 F 2 hypergeometric functions, with a simplification to the 2 F 1 hypergeometric function possible for p = 0 and p = 1 , allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence { M 2 p r } p = 0 ∞ . Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined. [ABSTRACT FROM AUTHOR]

In this paper, we study counting perfect matchings in linear chain graphs, focusing on identically colored and alternatingly colored odd faces, using recurrence relations. Our primary objective is to derive explicit formulas for the numbers of perfect matchings in linear chain graphs with identically colored odd faces. Furthermore, we establish a relationship between the numbers of perfect matchings in linear chain graphs with identically colored odd faces and strip snake chain graphs. This relationship provides us with an alternative way of validating the numbers of perfect matchings in linear chain graphs with the same colored odd faces. [ABSTRACT FROM AUTHOR]

In this article, we extend the classical framework for computing discriminants of special quasi-orthogonal polynomials from Schur's resultant formula, and establish a framework for computing discriminants of a sufficiently broader class of polynomials from the resultant formulas that are proven by Ulas and Turaj. More precisely, we derive a formula for the discriminant of a sequence { r A , n + c r A , n - 1 } of polynomials. Here, c is an element of a field K and { r A , n } is a sequence of polynomials satisfying a certain recurrence relation. There are several works computing the discriminants of given polynomials. For example, Kaneko–Niiho and Mahlburg–Ono independently proved the formula for the discriminants of certain hypergeometric polynomials that are related to j-invariants of supersingular elliptic curves. Sawa–Uchida proved the formula for the discriminants of quasi-Jacobi polynomials and applied it to prove the nonexistence of certain rational quadrature formulas. Our main theorem presents a uniform way to prove a vast generalization of the above formulas for the discriminants. [ABSTRACT FROM AUTHOR]

This work is intended to establish a relation between Vietoris’ sequence, which is a rational sequence, and hybrid numbers. Then it provides some characteristic properties of the hybrid numbers with Vietoris’ number coefficients. Some relations between this hybrid number and its norm, the recurrence relations, the generating function, Binet-like formula and Catalan-like identities are also indicated. Furthermore, a determinantal approach is presented to obtain elements of Vietoris’ hybrid number sequence. [ABSTRACT FROM AUTHOR]

Many matrix‐valued functions of an m×m Wishart matrix W, Fk(W), say, are homogeneous of degree k in W, and are equivariant under the conjugate action of the orthogonal group 풪(m), that is, Fk(HWHT)=HFk(W)HT, H∈풪(m). It is easy to see that the expectation of such a function is itself homogeneous of degree k in ∑, the covariance matrix, and are also equivariant under the action of 풪(m) on ∑. The space of such homogeneous, equivariant, matrix‐valued functions is spanned by elements of the type Wrpλ(W), where r∈{0,...,k} and, for each r, λ varies over the partitions of k−r, and pλ(W) denotes the power‐sum symmetric function indexed by λ. In the analogous case where W is replaced by W−1, these elements are replaced by W−rpλ(W−1). In this paper, we derive recurrence relations and analytical expressions for the expectations of such functions. Our results provide highly efficient methods for the computation of all such moments. [ABSTRACT FROM AUTHOR]

In this paper, a new generalization of Mersenne bihyperbolic numbers is introduced. Some of the properties of presented numbers are given. A general bilinear index-reduction formula for the generalized bihyperbolic Mersenne numbers is obtained. This result implies the Catalan, Cassini, Vajda, d'Ocagne and Halton identities. Moreover, generating function and matrix generators for these numbers are presented.

ABSTRACTIn this paper, we introduce general sequence of twice-iterated [Formula: see text]-(degenerate) Gould–Hopper Appell polynomials (TI-DGHAP) via discrete [Formula: see text]-Gould–Hopper Appell convolution. We obtain some of their characteristic properties such as explicit representation, determinantal representation, recurrence relation, lowering operator (LO), raising operator (RO), difference equation (DE), integro-partial lowering operator (IPLO), integro-partial raising operator (IPRO) and integro-partial difference equation (IPDE). As special cases of these general polynomials, we present TI degenerate Gould–Hopper Bernoulli polynomials, TI degenerate Gould–Hopper Poisson–Charlier polynomials, TI degenerate Gould–Hopper Boole polynomials and TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials. We also state their corresponding characteristic properties.

The algebraic composition of an element x of a commutative binary structure ℬ is modeled by its divisor (pseudo)graph Γx (which can be regarded as a simple graph if ℬ is an abelian group), whose vertices are the elements of ℬ such that two vertices a and b are adjacent if and only if ab = x. With respect to the action of a group of symmetries on the set of divisor pseudographs of ℬ, the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when ℬ is a finite abelian group. The minimum number of orbits is computed for every abelian group of order at most 11, and it is also determined for all finite cyclic groups. Moreover, systems of pseudographs that can be realized as those of finite abelian groups are completely characterized. In fact, recurrence relations are given for constructing systems of divisor pseudographs of finite cyclic groups, and all commutative binary structures that are isotopic to a finite abelian group are established. [ABSTRACT FROM AUTHOR]

In this study, we use discrete Appell convolution to define the sequence of Laguerre type twice-iterated Appell polynomials. We obtain explicit representation, recurrence relation, determinantal representation, lowering operator, integro-partial raising operator and integro-partial differential equation. In addition, the special cases of this new family are investigated using Euler and Bernoulli numbers. We also state their corresponding characteristic properties. [ABSTRACT FROM AUTHOR]

In this article, 3-parameter generalized quaternionic Leonardo p-sequence which is a quaternion generalization of the Leonardo p-sequence is presented. Several properties are examined by means of their homogeneous and non-homogeneous recursive relations. According to the special cases of 3-parameter, λ iє {1,2,3} and p all the results given in paper are reducible to all quaternions mentioned in this paper. [ABSTRACT FROM AUTHOR]

In this research, we introduce three novel sequences of γn, αn and βn. These sequences are related to each other through the recurrence relation, and we have observed that their relationship can be expressed using k-Fibonacci sequences. To prove this relationship, we used mathematical induction. We have shown the validity of our theorem, and the results are presented in this study. [ABSTRACT FROM AUTHOR]

In this paper we investigate a family of bivariate orthogonal functions arising as a generalization of Koornwinder polynomials in two variables. General properties like recurrence relations and partial differential equations are introduced. Some special cases are considered and a limit relation of these functions is studied. As a consequence, a new class of bivariate orthogonal polynomials is presented. [ABSTRACT FROM AUTHOR]

In this paper, the semi-local convergence of the Cordero's sixth-order iterative method in Banach space was proved by the method of recursion relation. In the process of proving, the auxiliary sequence and three increasing scalar functions can be derived using Lipschitz conditions on the first-order derivatives. By using the properties of auxiliary sequence and scalar function, it was proved that the iterative sequence obtained by the iterative method was a Cauchy sequence, then the convergence radius was obtained and its uniqueness was proven. Compared with Cordero's process of proving convergence, this paper does not need to ensure that G (s) is continuously differentiable in higher order, and only the first-order Fréchet derivative was used to prove semi-local convergence. Finally, the numerical results showed that the recursion relationship is reasonable. [ABSTRACT FROM AUTHOR]

In this paper, the semi-local convergence of the Cordero's sixth-order iterative method in Banach space was proved by the method of recursion relation. In the process of proving, the auxiliary sequence and three increasing scalar functions can be derived using Lipschitz conditions on the first-order derivatives. By using the properties of auxiliary sequence and scalar function, it was proved that the iterative sequence obtained by the iterative method was a Cauchy sequence, then the convergence radius was obtained and its uniqueness was proven. Compared with Cordero's process of proving convergence, this paper does not need to ensure that $ \mathcal{G}(s) $ is continuously differentiable in higher order, and only the first-order Fréchet derivative was used to prove semi-local convergence. Finally, the numerical results showed that the recursion relationship is reasonable.

In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the efficiency and applicability of the proposed algorithms.

The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented.

AbstractWe consider linear recursions of length two and related gap recursions where the indices may not be consecutive integers. Given a linear recursion of length two, we prove the existence of an explicit linear recursion of length [Formula: see text] with arbitrary distance between indices. Conversely, it is shown under some mild assumption that a linear recursion of length [Formula: see text] can be reduced to one of length two.

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices.

Connection between the interlacing of the zeros and the orthogonality of a given sequence of polynomials is done by K. Driver. In this paper, we attempt to extend this result to some particular cases of d-orthogonal polynomials. In fact, first, we characterize the 2-orthogonality of a given sequence $ \{ P_{n} \}_{n\geq 0} $ { P n } n ≥ 0 , with the existence of a certain ratio $ c_n $ c n expressed by means of the zeros of $ P_n $ P n . Then, for the $ (d+1) $ (d + 1) -fold symmetric polynomials, $ \{ P_{n} \}_{n\geq 0} $ { P n } n ≥ 0 , such that $ P_n $ P n has $ q_n $ q n distinct positive real zeros, $ n=(d+1)q_n+j, j=0,\ldots,d $ n = (d + 1) q n + j , j = 0 , ... , d , we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio $ c_n $ c n . Finally, we give necessary and sufficient conditions on the zeros of a given sequence $ \{ P_{n} \}_{n\geq 0} $ { P n } n ≥ 0 , that will assure that this sequence satisfies a particular $ (d+1) $ (d + 1) -order recurrence relation. Many examples to illustrate the obtained results are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce general sequence of twice-iterated Δw -(degenerate) Gould–Hopper Appell polynomials (TI-DGHAP) via discrete Δw -Gould–Hopper Appell convolution. We obtain some of their characteristic properties such as explicit representation, determinantal representation, recurrence relation, lowering operator (LO), raising operator (RO), difference equation (DE), integro-partial lowering operator (IPLO), integro-partial raising operator (IPRO) and integro-partial difference equation (IPDE). As special cases of these general polynomials, we present TI degenerate Gould–Hopper Bernoulli polynomials, TI degenerate Gould–Hopper Poisson–Charlier polynomials, TI degenerate Gould–Hopper Boole polynomials and TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials. We also state their corresponding characteristic properties. [ABSTRACT FROM AUTHOR]

In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the effciency and applicability of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of the affine space group Aff(3), considered with its natural action on the vector space of three-variable polynomials. The polynomial family {Bm,n,k (x, y, z)} is called quasi-monomial with respect to H if the group operators in two different bases {xmynzk} and {Bm,n,k (x, y, z)} have identical matrices. We derive a criterion for quasi-monomiality when the group H is the special orthogonal group SO(3). This criterion is expressed through the exponential generating function of the polynomial family {Bm,n,k (x, y, z)}. It has been proven that Appel’s biorthogonal polynomials are quasi-monomials with respect to SO(3) and recurrence relations have been found for them. [ABSTRACT FROM AUTHOR]

In this paper, a new generalization of Mersenne bihyperbolic numbers is introduced. Some of the properties of presented numbers are given. A general bilinear index-reduction formula for the generalized bihyperbolic Mersenne numbers is obtained. This result implies the Catalan, Cassini, Vajda, d'Ocagne and Halton identities. Moreover, generating function and matrix generators for these numbers are presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new hybrid number sequence with Horadam numbers and a finite operator, called Horadam finite operator hybrid numbers. We derive recurrence relation, Binet-like formula, ordinary generating function, exponential generating function, Poisson generating function, and summation formula for Horadam finite operator hybrid numbers. Moreover, we give matrix representation and Cassini's identities for these numbers. [ABSTRACT FROM AUTHOR]

We consider linear recursions of length two and related gap recursions where the indices may not be consecutive integers. Given a linear recursion of length two, we prove the existence of an explicit linear recursion of length l with arbitrary distance between indices. Conversely, it is shown under some mild assumption that a linear recursion of length l can be reduced to one of length two. [ABSTRACT FROM AUTHOR]

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. [ABSTRACT FROM AUTHOR]

We consider a confluent modification of the general Heun equation having three regular singular points and an irregular singularity of rank one at infinity. We show that this equation admits infinitely many solutions in terms of a single generalized confluent hypergeometric function. The solutions apply if two regular singularities have non-zero integer characteristic exponents and each of the two accessory parameters obeys a polynomial equation. [ABSTRACT FROM AUTHOR]

We study two families of type II discrete multiple orthogonal polynomials on an -legged star-like set with respect to weight functions of Charlier (Poisson distributions) and Meixner (negative binomial distributions), respectively. We focus our attention on the structural properties such as the raising operators, the Rodrigues-type formulas, and the explicit expressions of the polynomial families as well as the coefficients of the nearest neighbor recurrence relation. A limit relation between these families is given. [ABSTRACT FROM AUTHOR]

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals.

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics.

In this paper, the semilocal convergence of the eighth order iterative method is proved in Banach space by using the recursive relation, and the proof process does not need high order derivative. By selecting the appropriate initial point and applying the Lipschitz condition to the first order Fréchet derivative in the whole region, the existence and uniqueness domain are obtained. In addition, the theoretical results of semilocal convergence are applied to two nonlinear systems, and satisfactory results are obtained.

A recursive scheme was proposed to calculate two‐electron integrals of frequency‐dependent Breit interactions in electronic structure calculations using Gaussian basis functions. As shown in a previous study [R. Ahlrichs, Phys. Chem. Chem. Phys. 8 (2006) 3072–3077], the vertical recurrence relation for the two‐electron integrals of the general two‐body potential is valid. In addition, the authors have shown that the horizontal case is also valid. Explicit expressions for generalized molecular incomplete gamma function corresponding to the frequency‐dependent Gaunt and gauge potentials were then derived, along with their asymptotic formulas. In addition, an implementation for computing the generalized molecular incomplete gamma function was proposed. Through numerical calculations, the shape of the curves of the generalized molecular incomplete gamma functions were found to vary significantly from that of the zero‐energy case with the increase in the energy variable. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the study of finite sequences of orthogonal polynomials from one to two variables. In doing so, twenty three new classes of bivariate finite orthogonal polynomials are presented, obtained from the product of a finite and an infinite family of univariate orthogonal polynomials. For these new classes of bivariate finite orthogonal polynomials, we present a bivariate weight function, the domain of orthogonality, the orthogonality relation, the recurrence relations, the second-order partial differential equations, the generating functions, as well as the parameter derivatives. The limit relations among these families are also presented in Labelle's flavor. [ABSTRACT FROM AUTHOR]

In this paper, We present an algebraic theory of classical d-orthogonal polynomials and we want to fill in some gaps. We broaden and close some inclusions that exist and are known perhaps as consequences. Several characterizations of classical d-OPS are given in terms of d+2 term recurrence relation as well as in terms of functional and differential equations. An additional tool in determining the integral representation for such class is presented. [ABSTRACT FROM AUTHOR]

Let Nwm be the set of positive integers 1, 2,, · · ·, m and S ⊆ Nm. In 2000, J. Caumeran and R. Corcino made a thorough investigation on counting restricted functions f|S under each of the following conditions: (a) f(a) ≤ a, ∀a ∈ S; (b) f(a) ≤ g(a), ∀a ∈ S where g is any nonnegative real-valued continuous functions; (c) g1(a) ≤ f(a) ≤ g2(a), ∀a ∈ S, where g1 and g2 are any nonnegative real-valued continuous functions. Several formulae and identities were also obtain by Caumeran using basic concepts in combinatorics. In this paper we count those restricted functions under condition f(a) ≤ a, ∀a ∈ S which is one-to-one and onto and establish some formulas and identities parallel to those obtained by J. Caumeran and R. Corcino. [ABSTRACT FROM AUTHOR]

In this paper, we derive some classical and fractional properties of the r R s matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials. [ABSTRACT FROM AUTHOR]