generating function
در نشریات گروه ریاضی-
Given an integer $n\geq4$, how many inequivalent quadrilaterals with ordered integer sides and perimeter $n$ are there? Denoting such number by $Q(n)$, the answer is given by the following closed formula:\[Q(n)=\left\{ \dfrac{1}{576}n\left( n+3\right) \left( 2n+3\right) -\dfrac{\left( -1\right) ^{n}}{192}n\left( n-5\right) \right\} \cdot\]Keywords: Integer quadrilaterals, Ordered quadrilaterals, Integer partitions, generating function
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In this study we define a new generalization of the hybrid Leonardo sequence consisting of the hybrid numbers with hybrid Leonardo numbers coefficients. We investigate some algebraic properties of this new sequence and also the generating function, exponential generating function, and the Binet formula related to this type of sequence. In addition, some identities are provided, such as Catalans and Cassini’s identities, and sums are related.
Keywords: Leonardo Sequence, Hybrid Leonardo Sequence, The Binet Formula, Generating Function -
International Journal Of Nonlinear Analysis And Applications, Volume:16 Issue: 2, Feb 2025, PP 245 -253In this research, new representations of basic functions are proposed based on the new types of fuzzy partition and a subnormal generating function. The generalized uniform fuzzy partitions in subnormal case, i.e. in case a generating function K is not normal (generalized normal case), and simpler form of fuzzy transform (FzT) components based on these new representations of the generalized uniform fuzzy partitions are indicated. The main properties of a new uniform fuzzy partition are suggested. New theorems and lemmas are proved.Keywords: Fuzzy Partition, Fuzzy Transform, Basic Function, The Membership Functions, Generating Function, Ruspini Condition
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The aim of this paper is to introduce the hyperbolic generalized $k$-Horadam quaternions and octonions and investigate their algebraic properties. We present some properties and identities of these quaternions and octonions for generalized $k$-Horadam numbers. Moreover, we give some determinants related to the hyperbolic generalized $k$-Horadam quaternions and octonions. Finally, we evaluate its determinants through the Chebyshev polynomials of the second kind and give an illustrative example as well.Keywords: Horadam Number, Hyperbolic Quaternions, Octonions, Binet Formula, Generating Function, Chebyshev Polynomials
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We obtain the generating function for the number of columns of fixed height $r$ in a bargraph (classified according to semi-perimeter). As initial case for two distinct methods we first find the generating function for columns of height $1$. Then using a first-return-to-level-$1$ decomposition, we obtain the rational function version of the continued fraction generating function which allows us to derive separate recursions for its numerator and denominator. This then allows us to get the asymptotic average number of columns for each $r$. We also obtain an equivalent generating function by exploiting a sequential decomposition for bargraphs in terms of columns of height $r$.Keywords: generating function, bargraphs, column height
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The continuous dual Hahn polynomials are orthogonal polynomials in a single variable whose weight function is given by the product of the gamma function. In this paper, we derive some advanced properties for these polynomials including multilinear and multilateral generating functions, recurrence relations and various integral representations.
Keywords: Hypergeometric Series, Continuous DualHahn Polynomials, Generating Function, Recurrence Relation, IntegralRepresentation -
International Journal Of Nonlinear Analysis And Applications, Volume:14 Issue: 1, Jan 2023, PP 2123 -2128We propose a beta exponential distribution that is formed from the logarithm of a random variable with a beta value, as well as a thorough investigation of the distribution's mathematical characteristics. Because we dealt with the beta exponential distribution, we have a clear and understandable way to comprehend the equation and apply it to the actual problem. To do this, we gathered the questions that statisticians find interesting and studied the most significant properties and statistics related to the distribution. Future research will utilize this data to identify specific industrial flaws or inefficiencies as a function of survival. Included are system statistics, BE distributions for graphs, generation functions, moments, and Hazard functions.Keywords: Hazard function, Moment, Generating Function, shape, Order statistics of BE Distribution
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In this study, a generalization of the Pell sequence called bi-periodic Pell sequence is carried out to matrix theory. Therefore, we call this matrix sequence the bi-periodic Pell matrix sequence whose entries are bi-periodic Pell numbers. Then the generating function, Binet formula and some basic properties and sum formulas are examined.
Keywords: Pell Sequence, Generating Function, Binet Formula -
A non-empty set S ⊆ V is a dominating set, if every vertex not in S is adjacent to at least one vertex in S, and S is a total dominating set, if every vertex of V is adjacent to some vertices of S. We enumerate dominating sets, non-split dominating sets and total dominating sets in several classes of cactus chains.Keywords: Dominating sets, Total dominating sets, Generating function, Cactus graphs, i-uniform
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We investigate the eigenvalue distribution of banded Hankel matrices with non-zero skew diagonals. This work uses push-forward of an arcsine density, block structures and generating functions. Our analysis is done by a combination of Chebyshev polynomials, Laplacian determinant expansion and mathematical induction.Keywords: Hankel, eigenvalue, Distribution, generating function
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In this work we study numbers and polynomials generated by two type of composition of generating functions and get their explicit formulae. Furthermore we state an improvementof the composita formulae's given in [6] and [3], using the new composita formula's we construct a variety of combinatorics identities. This study go alone to dene new family of generalized Bernoulli polynomials which include Hermite-Bernoulli polynomials introduced by G. Dattoli and al [1].Keywords: Generating function, composition of generating func- tions, composita, Faa di Bruno formula
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We find an explicit expression of the Tutte polynomial of an $n$-fan. We also find a formula of the Tutte polynomial of an $n$-wheel in terms of the Tutte polynomial of $n$-fans. Finally, we give an alternative expression of the Tutte polynomial of an $n$-wheel and then prove the explicit formula for the Tutte polynomial of an $n$-wheel.Keywords: Tutte polynomial, wheel, fan, generating function
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Bentea and Tu{a}rnu{a}uceanu~(An. c{S}tiinc{t}. Univ. Al. I. Cuza Iac{s}, Ser. Nouv{a}, Mat., {bf 54(1)} (2008), 209-220) proposed the following problem: Find an explicit formula for the number of fuzzy subgroups of a finite hamiltonian group of type $Q_8times mathbb{Z}_n$ where $Q_8$ is the quaternion group of order $8$ and $n$ is an arbitrary odd integer. In this paper we consider more general group: the direct product of a generalized quaternion group of any even order and a cyclic group of any odd order. For this group we give an explicit formula for the number of fuzzy subgroups.Keywords: Generalized quaternion group, Hamiltonian group, Fuzzy subgroups, Subgroup chain, Generating function
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Let n β(n) o∞ n=0 be a sequence of positive numbers and 1 < p < ∞. We consider the weighted Hardy space Hp (β). We investigate the relation between the generating function and the functional of point evaluations. Also, under a sufficient condition we determine the structure of all non-zero multiplicative linear functionals on Hp (β).
Keywords: The Banach space of formal powerseries associated with a sequence β, bounded point evaluation, generating function
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