1,720,979 research outputs found
-Hypergeometric Orthogonal Polynomials with = −1
No full textWe obtain some properties of a class of -hypergeometric orthogonal polynomials with = −1, described by a uniform parametrization of the recurrence coefficients. We construct a class of complementary −1 polynomials by means of the Darboux transformation with a shift. We show that our classes contain the Bannai-Ito polynomials, their complementary polynomials, and other known −1 polynomials. We introduce some new examples of −1 polynomials and also obtain matrix realizations of the Bannai-Ito algebra.I am very grateful to the anonymous referees for their many suggestions, which helped improve the paper
Polynomial sequences generated by infinite Hessenberg matrices
No full textWe show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix
Linearization and connection coefficients of polynomial sequences: A matrix approach
Get PDFFor a sequence of polynomials in one real or complex variable,
where has degree , for , we find explicit expressions and
recurrence relations for infinite matrices whose entries are the coefficients
, called linearization coefficients, that satisfy
For any pair of polynomial sequences and we find
infinite matrices whose entries are the coefficients that satisfy
Such results are obtained using a matrix approach. We also obtain recurrence
relations for the linearization coefficients, apply the general results to
general orthogonal polynomial sequences and to particular families of
orthogonal polynomials such as the Chebyshev, Hermite, and Charlier families.Comment: 14 page
Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices
No full textWe use elementary triangular matrices to obtain some factorization, multiplication, and inversion
properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg
matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg
matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the
identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n
On linear matrix differential equations
No full textAbstractWe use elementary methods and operator identities to solve linear matrix differential equations and we obtain explicit formulas for the exponential of a matrix. We also give explicit constructions of solutions of scalar homogeneous equations with certain initial values, called dynamic solutions, that play an important role in the solution of homogeneous and non-homogeneous matrix differential equations. We show that the same methods can be used to solve linear matrix difference equations
Functions of matrices
No full textAbstractWe use the Cayley–Hamilton theorem and the sequence of Horner polynomials associated with a polynomial w(z) to obtain explicit formulas for functions of the form f(tA), where f is defined by a convergent power series and A is a square matrix. We use a well-known explicit formula for the resolvent of A and show that f(tA) is the Hadamard product of f(t) and the function (I−tA)−1, which is easily obtained from the resolvent
A Linear Algebra approach to monomiality and operational methods
Get PDFWe use linear algebraic methods to obtain general results about linear
operators on a space of polynomials that we apply to the operators associated
with a polynomial sequence by the monomiality property. We show that all such
operators are differential operators with polynomial coefficients of finite of
infinite order. We consider the monomiality operators associated with several
classes of polynomial sequences, such as Appell and Sheffer, and also
orthogonal polynomial sequences that include the Meixner, Krawtchouk, Laguerre,
Meixner-Pollaczek, and Hermite families.Comment: 28 page
- …
