On generalized Laguerre matrix polynomials

Abstract The main object of the present paper is to introduce and study the generalized Laguerre matrix polynomials for a matrix that satisfies an appropriate spectral property. We prove that these matrix polynomials are characterized by the generalized hypergeometric matrix function. An explicit representation, integral expression and some recurrence relations in particular the three terms recurrence relation are obtained here. Moreover, these matrix polynomials appear as solution of a differential equation.

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Some new results for Jacobi matrix polynomials

Filomat ◽

10.2298/fil1304713c ◽

2013 ◽

Vol 27 (4) ◽

pp. 713-719 ◽

Cited By ~ 2

Author(s):

Bayram Çekim ◽

Abdullah Altin ◽

Rabia Aktaş

Keyword(s):

Matrix Function ◽

Recurrence Relations ◽

Jacobi Matrix ◽

Integral Representations ◽

Matrix Polynomials ◽

Generating Matrix

The main aim of this paper is to obtain some recurrence relations and generating matrix function for Jacobi matrix polynomials (JMP). Also, various integral representations satisfied by JMP are derived.

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On Lommel Matrix Polynomials

Symmetry ◽

10.3390/sym13122335 ◽

2021 ◽

Vol 13 (12) ◽

pp. 2335

Author(s):

Ayman Shehata

Keyword(s):

Entire Function ◽

Recurrence Relations ◽

Complex Analysis ◽

Order Type ◽

Explicit Representation ◽

Integral Representations ◽

Matrix Polynomials ◽

New Class ◽

Special Cases ◽

Matrix Recurrence Relations

The main aim of this paper is to introduce a new class of Lommel matrix polynomials with the help of hypergeometric matrix function within complex analysis. We derive several properties such as an entire function, order, type, matrix recurrence relations, differential equation and integral representations for Lommel matrix polynomials and discuss its various special cases. Finally, we establish an entire function, order, type, explicit representation and several properties of modified Lommel matrix polynomials. There are also several unique examples of our comprehensive results constructed.

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A class of big (p,q)-Appell polynomials and their associated difference equations

Filomat ◽

10.2298/fil1910085s ◽

2019 ◽

Vol 33 (10) ◽

pp. 3085-3121

Author(s):

H.M. Srivastava ◽

B.Y. Yaşar ◽

M.A. Özarslan

Keyword(s):

Differential Equation ◽

Difference Equation ◽

Recurrence Relation ◽

Difference Equations ◽

Recurrence Relations ◽

Bernoulli Polynomials ◽

Equivalence Theorem ◽

Euler Polynomials ◽

Appell Polynomials ◽

Special Case

In the present paper, we introduce and investigate the big (p,q)-Appell polynomials. We prove an equivalance theorem satisfied by the big (p, q)-Appell polynomials. As a special case of the big (p,q)- Appell polynomials, we present the corresponding equivalence theorem, recurrence relation and difference equation for the big q-Appell polynomials. We also present the equivalence theorem, recurrence relation and differential equation for the usual Appell polynomials. Moreover, for the big (p; q)-Bernoulli polynomials and the big (p; q)-Euler polynomials, we obtain recurrence relations and difference equations. In the special case when p = 1, we obtain recurrence relations and difference equations which are satisfied by the big q-Bernoulli polynomials and the big q-Euler polynomials. In the case when p = 1 and q ? 1-, the big (p,q)-Appell polynomials reduce to the usual Appell polynomials. Therefore, the recurrence relation and the difference equation obtained for the big (p; q)-Appell polynomials coincide with the recurrence relation and differential equation satisfied by the usual Appell polynomials. In the last section, we have chosen to also point out some obvious connections between the (p; q)-analysis and the classical q-analysis, which would show rather clearly that, in most cases, the transition from a known q-result to the corresponding (p,q)-result is fairly straightforward.

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DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

Modern Physics Letters B ◽

10.1142/s021798490701244x ◽

2007 ◽

Vol 21 (02n03) ◽

pp. 139-154 ◽

Cited By ~ 6

Author(s):

J. H. ASAD

Keyword(s):

Differential Equation ◽

Green’S Function ◽

Recurrence Relation ◽

Green's Function ◽

Two Dimensional ◽

Dimensional Lattice ◽

One Dimensional ◽

The One ◽

Lattice Green's Function ◽

Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

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ELEMENTARY OSCILLATION CRITERIA FOR A THREE TERM RECURRENCE RELATION WITH OSCILLATORY COEFFICIENT SEQUENCE

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.29.1998.4274 ◽

1998 ◽

Vol 29 (3) ◽

pp. 227-232

Author(s):

GUANG ZHANG ◽

SUI-SUN CHENG

Keyword(s):

Recurrence Relation ◽

Recurrence Relations ◽

Qualitative Properties ◽

Oscillation Criteria ◽

Three Term Recurrence Relation

Qualitative properties of recurrence relations with coefficients taking on both positive and negative values are difficult to obtain since mathematical tools are scarce. In this note we start from scratch and obtain a number of oscillation criteria for one such relation : $x_{n+1}-x_n+p_nx_{n-r}\le 0$.

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On the quotient moment of lower generalized order statistics and characterization

Journal of Applied Mathematics Statistics and Informatics ◽

10.1515/jamsi-2015-0005 ◽

2015 ◽

Vol 11 (1) ◽

pp. 73-89

Author(s):

Devendra Kumar

Keyword(s):

Order Statistics ◽

Recurrence Relation ◽

Conditional Expectation ◽

General Class ◽

Recurrence Relations ◽

Generalized Order Statistics ◽

Lower Records ◽

Moments Of Order Statistics ◽

Generalized Order

Abstract In this paper we consider general class of distribution. Recurrence relations satisfied by the quotient moments and conditional quotient moments of lower generalized order statistics for a general class of distribution are derived. Further the results are deduced for quotient moments of order statistics and lower records and characterization of this distribution by considering the recurrence relation of conditional expectation for general class of distribution satisfied by the quotient moment of the lower generalized order statistics.

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Distinct partitions and overpartitions

Carpathian Journal of Mathematics ◽

10.37193/cjm.2022.01.12 ◽

2021 ◽

Vol 38 (1) ◽

pp. 149-158

Author(s):

MIRCEA MERCA ◽

Keyword(s):

Partition Function ◽

Recurrence Relation ◽

Recurrence Relations ◽

Convergent Series ◽

The Other ◽

Large Integer ◽

Linear Recurrence ◽

Fast Computation ◽

Real Numbers ◽

Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

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BOUNDARY VALUE PROBLEM WITH SHIFT FOR A FRACTIONAL ORDER DELAY DIFFERENTIAL EQUATION

Вестник КРАУНЦ. Физико-математические науки ◽

10.26117/2079-6641-2019-28-3-16-25 ◽

2019 ◽

pp. 16-25

Author(s):

М.Г. Мажгихова

Keyword(s):

Differential Equation ◽

Boundary Value Problem ◽

Fractional Order ◽

Delay Differential Equation ◽

Solvability Condition ◽

Boundary Value ◽

Explicit Representation ◽

Existence And Uniqueness Theorem ◽

The Green Function ◽

Delay Differential

В работе доказана теорема существования и единственности решения краевой задачи со смещением для дифференциального уравнения дробного порядка с запаздывающим аргументом. Решение задачи выписано в терминах функции Грина. Получено условие однозначной разрешимости и показано, что оно может нарушаться только конечное число раз. In this paper we prove existence and uniqueness theorem to a boundary value problem with shift for a fractional order ordinary delay differential equation. The solution of the problem is written out in terms of the Green function. We find an explicit representation for solvability condition and show that it may only be violated a finite number of times

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An Analogue of a Problem of J. Balázs and P. Turán

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-006-2 ◽

1969 ◽

Vol 21 ◽

pp. 54-63 ◽

Cited By ~ 5

Author(s):

J. Prasad ◽

A. K. Varma

Keyword(s):

Differential Equation ◽

Modulus Of Continuity ◽

Legendre Polynomial ◽

Existence And Uniqueness ◽

Explicit Representation ◽

Lacunary Interpolation ◽

First Derivative ◽

Second Derivatives ◽

Significant Application

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.

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