In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

Orthogonal Polynomials

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

/pdf/the-askey-scheme-of-hypergeometric-orthogonal-polynomials-46afheftah.pdf

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

We present quantitative optical images of human breast in vivo. The images were obtained by using near-infrared diffuse optical tomography (DOT) after the administration of indocyanine green (ICG) for contrast enhancement. The optical examination was performed concurrently with a magnetic resonance imaging (MRI) exam on patients scheduled for excisional biopsy or surgery so that accurate image coregistration and histopathological information of the suspicious lesions was available. The ICG-enhanced optical images coregistered accurately with Gadolinium-enhanced magnetic resonance images validating the ability of DOT to image breast tissue. In contrast to simple transillumination, we found that DOT provides for localization and quantification of exogenous tissue chromophore concentrations. Additionally our use of ICG, an albumin bound absorbing dye in plasma, demonstrates the potential to differentiate disease based on the quantified enhancement of suspicious lesions.

/pdf/concurrent-mri-and-diffuse-optical-tomography-of-breast-2tse322iji.pdf

Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement

A method for reconstructing images from projections is described The unique aspect of the procedure is that the reconstruction of the internal structure can be carried out for objects that diffuse the incident radiation The method may be used with photons, phonons, neutrons, and many other kinds of radiation The procedure has applications to medical imaging, industrial imaging, and geophysical imaging

Image Reconstruction of the Interior of Bodies That Diffuse Radiation

We develop a general method that allows us to introduce families of orthogonal matrix polynomials of size N × N satisfying second-order differential equations. The presence of this extra property should make these orthogonal polynomials into useful tools in several areas of mathematics and its applications. Historically, this has certainly been the case for their scalar-valued versions. The subtlety of the noncommutative algebra of matrices can be exploited to yield many different such families, almost dwarfing the scalar situation by comparison. All these families form a nice and rich hierarchy starting from the classical Jacobi, Hermite, and Laguerre families, N=1, and increasing in number and variety as the size N increases. We illustrate the use of our method by giving large classes of generic examples of arbitrary size N.

Orthogonal matrix polynomials satisfying second-order differential equations

http://directory.umm.ac.id/Data%20Elmu/jurnal/J-a/Journal%20of%20Computational%20And%20Applied%20Mathematics/Vol106.Issue2.1999/2785.pdf

F. Alberto Grünbaum

Papers

Image Reconstruction of the Interior of Bodies That Diffuse Radiation

Orthogonal matrix polynomials satisfying second-order differential equations

Some functions that generalize the Krall-Laguerre polynomials

The eigenvectors of the discrete Fourier transform: A version of the Hermite functions

The q -version of a theorem of Bochner