which are also orthogonal

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1: 18.2 General Orthogonal Polynomials

… ►In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system) …

2: 18.30 Associated OP’s

… ►Orthogonality … ►Orthogonality … ►They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10). ►Orthogonality … ►

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. …One of his most influential papers Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials (with J. …Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. … ►Additional books for which Askey served as author or editor include Orthogonal Polynomials and Special Functions, published by SIAM in 1975, Theory and application of special functions, published by Academic Press in 1975, Special Functions: Group Theoretical Aspects and Applications (with T. …

4: 18.42 Software

… ►Also included is a website (CAOP) operated by a university department. Citations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. … ►

•

CAOP (website). Computer Algebra and Orthogonal Polynomials.

…

5: 2.8 Differential Equations with a Parameter

… ►in which

u

is a real or complex parameter, and asymptotic solutions are needed for large

|u|

that are uniform with respect to

z

in a point set

\mathbf{D}

\mathbb{R}

\mathbb{C}

. For example,

u

can be the order of a Bessel function or degree of an orthogonal polynomial. The form of the asymptotic expansion depends on the nature of the transition points in

\mathbf{D}

, that is, points at which

f(z)

has a zero or singularity. Zeros of

f(z)

are also called turning points. … ►The approach used in preceding subsections for equation (2.8.1) also succeeds when

z_{0}

is a multiple or fractional turning point. …

6: 18.41 Tables

… ►For

P_{n}\left(x\right)

(

=\mathsf{P}_{n}\left(x\right)

) see §14.33. … ►The precision is 10D, except for

H_{n}\left(x\right)

which is 6-11S. … ►See also Abramowitz and Stegun (1964, Tables 25.4, 25.9, and 25.10). …

7: Bille C. Carlson

… ►Navy after the onset of World War II and worked on the island of Guam with radar, which was novel at the time. … ►In theoretical physics he is known for the “Carlson-Keller Orthogonalization”, published in 1957, Orthogonalization Procedures and the Localization of Wannier Functions, and the “Carlson-Keller Theorem”, published in 1961, Eigenvalues of Density Matrices. … ►This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. …Also, the homogeneity of the

R

-function has led to a new type of mean value for several variables, accompanied by various inequalities. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. …

8: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►Taken together, these expansions are uniformly valid for

-\infty<x<\infty

and for

a

in unbounded intervals—each of which contains

[0,(1-\delta)n]

, where

\delta

again denotes an arbitrary small positive constant. See also Bo and Wong (1994) and Goh (1998). … ►Compare also (18.21.12). Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

9: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

… ► … ►

Orthogonality

… ►These polynomials are also called Rogers polynomials. … ►For continuous

q^{-1}

-Hermite polynomials the orthogonality measure is not unique. …

10: 18.27 $q$ -Hahn Class

… ►

§18.27(i) Introduction

… ►All these systems of OP’s have orthogonality properties of the form …Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … ►The orthogonality relations are given by (18.27.2), with … ►

Limit Relations

…