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1: Alexander I. Bobenko

… ► Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …He is also coeditor of Discrete Integrable Geometry and Physics (with R.  Seiler), published by Oxford University Press in 1999, and Discrete Differential Geometry (with P. …

2: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

3: 26.22 Software

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GAP (website). A system for computational discrete algebra.

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4: 20.11 Generalizations and Analogs

… ►It is a discrete analog of theta functions. If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …This is the discrete analog of the Poisson identity (§1.8(iv)). …

5: 18.27 $q$-Hahn Class

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§18.27(vii) Discrete $q$-Hermite I and II Polynomials

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Discrete $q$-Hermite I

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Discrete $q$-Hermite II

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18.27.24 $$\sum _{\mathrm{\ell }=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\tilde{h}}_n(c{q}^{\mathrm{\ell }};q){\tilde{h}}_m(c{q}^{\mathrm{\ell }};q)+{\tilde{h}}_n(-c{q}^{\mathrm{\ell }};q){\tilde{h}}_m(-c{q}^{\mathrm{\ell }};q)\right)\frac{q^{\mathrm{\ell }}}{{(-c^2{q}^{2\mathrm{\ell }};q^2)}_{\mathrm{\infty }}}=2\frac{{(q^2,-c^{2}q,-c^{-2}q;q^2)}_{\mathrm{\infty }}}{{(q,-c^2,-c^{-2}{q}^2;q^2)}_{\mathrm{\infty }}}\frac{{(q;q)}_n}{q^{n^2}}{\delta }_{n,m},$$ $c>0$.

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${\tilde{h}}_n(x;q)$: discrete $q$-Hermite II polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.27.E24

Encodings:

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See also:

Annotations for §18.27(vii), §18.27(vii), §18.27 and Ch.18

►For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …

6: Bibliography Y

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$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

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Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

Encodings:

BibTeX

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7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► The analogous orthonormality is … ►These sets may be discrete, continuous, or a combination of both, as discussed in the following three subsections. … ►

§1.18(v) Point Spectra and Eigenfunction Expansions

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8: 18.39 Applications in the Physical Sciences

… ►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. … ►However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of $\mathscr{H}$ form a discrete, normed, and complete basis for a Hilbert space. … ►The spectrum is entirely discrete as in §1.18(v). … ►The spectrum is entirely discrete as in §1.18(v). … ►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified: …

9: 18.1 Notation

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Discrete $q$-Hermite I: $h_n(x;q)$.

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Discrete $q$-Hermite II: ${\tilde{h}}_n(x;q)$.

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10: 18.19 Hahn Class: Definitions

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Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

… ►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). … ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

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	$p_n(x)$	$X$	$w_x$	$h_n$
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