discrete analog

SearchAdvancedHelp

(0.001 seconds)

11—20 of 85 matching pages

11: 25.17 Physical Applications

… ►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. …

12: 26.22 Software

… ►

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

…

13: 5.16 Sums

… ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

14: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

… ►

$q$-Analog of Bailey’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

… ►

$q$-Analog of Gauss’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

… ►

F. H. Jackson’s Terminating $q$-Analog of Dixon’s Sum

… ►

$q$-Analog of Dixon’s ${}_3{}^{}F_2^{}\left(1\right)$ Sum

… ►

Gasper–Rahman $q$-Analogs of the Karlsson–Minton Sums

…

15: 18.27 $q$-Hahn Class

… ►

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

►

Discrete $q$-Hermite I

… ►

Discrete $q$-Hermite II

… ►

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

ⓘ

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:

pMML, png, TeX

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. …

16: 7.22 Methods of Computation

… ►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. …

17: Bibliography Y

… ►

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

ⓘ

Referenced by:

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

Encodings:

BibTeX

…

18: 28.30 Expansions in Series of Eigenfunctions

… ►For analogous results to those of §28.19, see Schäfke (1960, 1961b), and Meixner et al. (1980, §1.1.11).

19: Bille C. Carlson

… ►In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …

20: 6.4 Analytic Continuation

… ►

6.4.4 $$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

…