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.

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

We study a certain random growth 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 (GUE).

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Shape fluctuations and random matrices

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

At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, the exponential function, the probability integral and so on. Elliptic integrals proved to be the most important. They are connected with rectification of arcs of certain curves. The remarkable idea of Abel to replace these integrals by the corresponding inverse functions led to the creation of the theory of elliptic functions.

Representation of Lie groups and special functions

This paper introduces a new set of orthogonal moment functions based on the discrete Tchebichef polynomials. The Tchebichef moments can be effectively used as pattern features in the analysis of two-dimensional images. The implementation of the moments proposed in this paper does not involve any numerical approximation, since the basis set is orthogonal in the discrete domain of the image coordinate space. This property makes Tchebichef moments superior to the conventional orthogonal moments such as Legendre moments and Zernike moments, in terms of preserving the analytical properties needed to ensure information redundancy in a moment set. The paper also details the various computational aspects of Tchebichef moments and demonstrates their feature representation capability using the method of image reconstruction.

Image analysis by Tchebichef moments

With students of Physics chiefly in mind, we have collected the material on special functions that is most important in mathematical physics and quan tum mechanics We have not attempted to provide the most extensive collec tion possible of information about special functions, but have set ourselves the task of finding an exposition which, based on a unified approach, ensures the possibility of applying the theory in other natural sciences, since it pro vides a simple and effective method for the independent solution of problems that arise in practice in physics, engineering and mathematics For the American edition we have been able to improve a number of proofs; in particular, we have given a new proof of the basic theorem ( 3) This is the fundamental theorem of the book; it has now been extended to cover difference equations of hypergeometric type ( 12, 13) Several sections have been simplified and contain new material We believe that this is the first time that the theory of classical or thogonal polynomials of a discrete variable on both uniform and nonuniform lattices has been given such a coherent presentation, together with its various applications in physics"

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Special functions of mathematical physics : a unified introduction with applications

The basic properties of the polynomials p n (x) that satisfy the orthogonality relations 

$$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m \ne n) $$

(2.0.1)

hold also for the polynomials that satisfy the orthogonality relations of a more general form, which can be expressed in terms of Stielties integrals 

$$ \int_a^b {{p_n}(x)} {p_m}(x)dw(x) = 0\quad (m \ne n), $$

(2.0.2)

where w(x) is a monotonic nondecreasing function (usually called the distribution function). The orthogonality relation (2.0.2) is reduced to (2.0.1) in the case when the function w(x) has a derivative on (a, b) and w′(x) = ϱ(x). For solving many problems orthogonal polynomials are used that satisfy the orthogonality relations (2.0.2) in the case when w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ i at the points x = x i . In this case the orthogonality relation (2.0.2) can be rewritten in the form 

$$ \sum\limits_i {{p_n}({x_i})pm} ({x_i}){\rho_i} = 0\quad (m \ne n). $$

(2.0.3)

Classical Orthogonal Polynomials of a Discrete Variable

Many important problems of theoretical and mathematical physics lead to the differential equation 

$$u'' + \frac{{\tilde \tau (z)}}{{\sigma (z)}}u' + \frac{{\tilde \sigma (z)}}{{{\sigma ^2}(z)}}u = 0$$

(1)

where σ(z) and $\tilde \sigma (z)$ are polynomials of degree at most 2, and $\tilde \tau (z)$ is a polynomial of degree at most 1. Equations of this form arise, for example, in solving the Laplace and Helmholtz equations in curvilinear coordinate systems by the method of separation of variables, and in the discussion of such fundamental problems of quantum mechanics as the motion of a particle in a spherically symmetric field, the harmonic oscillator, the solution of the Schrodinger, Dirac and Klein-Gordon equations for a Coulomb potential, and the motion of a particle in a homogeneous electric or magnetic field. Moreover, equation (1) also arises in typical problems of atomic, molecular and nuclear physics.

Arnold F. Nikiforov

Papers

Special functions of mathematical physics : a unified introduction with applications

Classical Orthogonal Polynomials of a Discrete Variable

Foundations of the Theory of Special Functions

Classical orthogonal polynomials of A discrete variable and representations of the three-dimensional rotation group

Isothermal decay of thermoluminescence and energy distribution of traps in Al2O3–BeO ceramics