On Sieved Orthogonal Polynomials. V: Sieved Pollaczek Polynomials

AbstractWe examine the convergence and analytic properties of a continued fraction of Ramanujan and its connection to the orthogonal polynomials of Meixner-Pollaczek.

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Turán inequalities for symmetric orthogonal polynomials

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117129700001x ◽

1997 ◽

Vol 20 (1) ◽

pp. 1-7 ◽

Cited By ~ 5

Author(s):

Joaquin Bustoz ◽

Mourad E. H. Ismail

Keyword(s):

Orthogonal Polynomials ◽

Recurrence Relation ◽

Bessel Functions ◽

Sum Of Squares ◽

Weighted Sum ◽

Pollaczek Polynomials ◽

Lommel Polynomials ◽

Three Term Recurrence Relation

A method is outlined to express a Turán determinant of solutions of a three term recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Turán determinants of symmetric Pollaczek polynomials, Lommel polynomials andq-Bessel functions.

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THE CONSTRUCTION OF SUBORDINATED PROBABILITY MEASURES ON ℂ ASSOCIATED WITH THE JACOBI–SZEGÖ PARAMETERS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025709003823 ◽

2009 ◽

Vol 12 (03) ◽

pp. 401-411 ◽

Cited By ~ 2

Author(s):

NOBUHIRO ASAI

Keyword(s):

Probability Measure ◽

Orthogonal Polynomials ◽

Mellin Transform ◽

Bessel Functions ◽

Probability Measures ◽

Modified Bessel Functions ◽

Hahn Polynomials ◽

Pollaczek Polynomials

In this paper, it will be shown that a probability measure on ℂ associated with the Jacobi–Szegö parameters of the orthogonal polynomials can be obtained by making use of the classical Mellin transform and its convolution property. We shall construct several measures on ℂ represented by the modified Bessel functions. The material in this paper gives nontrivial examples originated from the continuous dual Hahn polynomials (one of hypergeometric orthogonal polynomials), which are beyond the Meixner–Pollaczek polynomials appeared in our previous papers.4, 5

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On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

Mathematics ◽

10.3390/math9090955 ◽

2021 ◽

Vol 9 (9) ◽

pp. 955

Author(s):

Abey S. Kelil ◽

Alta S. Jooste ◽

Appanah R. Appadu

Keyword(s):

Integral Representation ◽

Weight Function ◽

Orthogonal Polynomials ◽

Finite Order ◽

Recurrence Relations ◽

Pollaczek Polynomials ◽

Perturbed Polynomials ◽

Recursive Relations ◽

Fisher's Information

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.

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