147 research outputs found
Tur\'an type inequalities for Tricomi confluent hypergeometric functions
Some sharp two-sided Tur\'an type inequalities for parabolic cylinder
functions and Tricomi confluent hypergeometric functions are deduced. The
proofs are based on integral representations for quotients of parabolic
cylinder functions and Tricomi confluent hypergeometric functions, which arise
in the study of the infinite divisibility of the Fisher-Snedecor F
distribution. Moroever, some complete monotonicity results are given concerning
Tur\'an determinants of Tricomi confluent hypergeometric functions. These
complement and improve some of the results of Ismail and Laforgia [23].Comment: 20 page
Spectral Analysis of Certain Schr\"odinger Operators
The -matrix method is extended to difference and -difference operators
and is applied to several explicit differential, difference, -difference and
second order Askey-Wilson type operators. The spectrum and the spectral
measures are discussed in each case and the corresponding eigenfunction
expansion is written down explicitly in most cases. In some cases we encounter
new orthogonal polynomials with explicit three term recurrence relations where
nothing is known about their explicit representations or orthogonality
measures. Each model we analyze is a discrete quantum mechanical model in the
sense of Odake and Sasaki [J. Phys. A: Math. Theor. 44 (2011), 353001, 47
pages]
Classes of Bivariate Orthogonal Polynomials
We introduce a class of orthogonal polynomials in two variables which
generalizes the disc polynomials and the 2- Hermite polynomials. We identify
certain interesting members of this class including a one variable
generalization of the 2- Hermite polynomials and a two variable extension of
the Zernike or disc polynomials. We also give -analogues of all these
extensions. In each case in addition to generating functions and three term
recursions we provide raising and lowering operators and show that the
polynomials are eigenfunctions of second-order partial differential or
-difference operators
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