connection with orthogonal polynomials on the line

SearchAdvancedHelp

(0.001 seconds)

31—40 of 400 matching pages

31: 18.1 Notation

… ►

Classical OP’s

… ►

Hahn Class OP’s

… ►

Wilson Class OP’s

… ►

Disk: $R_{m,n}^{(\alpha )}\left(z\right)$.

►

Triangle: $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$.

…

32: 36.5 Stokes Sets

… ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. … ►This part of the Stokes set connects two complex saddles. … ►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …

33: 32.15 Orthogonal Polynomials

§32.15 Orthogonal Polynomials

►Let $p_n(\xi )$, $n=0,1,\mathrm{\dots }$, be the orthonormal set of polynomials defined by ►

32.15.1 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{4}{\xi }^4-z{\xi }^2\right)p_m(\xi )p_n(\xi )d\xi ={\delta }_{m,n},$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $m$: integer, $n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

… ►

32.15.2 $$a_{n+1}(z)p_{n+1}(\xi )=\xi p_n(\xi )-a_n(z)p_{n-1}(\xi ),$$

ⓘ

Symbols:

$n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

… ►

34: 21.10 Methods of Computation

… ►

§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

… ►

Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

35: 28.27 Addition Theorems

… ►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

36: 8.22 Mathematical Applications

… ►

§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

►The function $\mathrm{\Gamma }(a,z)$, with $|\mathrm{ph}a|\le \frac{1}{2}\pi$ and $\mathrm{ph}z=\frac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)) on the critical line $\mathrm{\Re }s=\frac{1}{2}$. …

37: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

38: 18.41 Tables

… ►

§18.41(i) Polynomials

►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_n\left(x\right)$ and $U_n\left(x\right)$, and $0.5,1,3,5,10$ for $L_n\left(x\right)$ and $H_n\left(x\right)$. … ►For $P_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ see §3.5(v). …

39: Bibliography M

… ►

Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

ⓘ

External Links:

ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

►

Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).

ⓘ

External Links:

ISSN 1286-4889, FullText, MathReview, ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

►

Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 0-521-82472-9, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

… ►

Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

ⓘ

External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

Encodings:

BibTeX

… ►

Exceptional orthogonal polynomials.

ⓘ

Notes:

Lecture at ICMAT School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, available at https://www.icmat.es/RT/optrim/school/lectures/Milson/icmat17.pdf

Referenced by:

§18.36(vi), §18.38(iii).

Encodings:

BibTeX

…

40: 18.34 Bessel Polynomials

… ►

§18.34(ii) Orthogonality

… ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : … ►Orthogonality of the full system on the unit circle can be given with a much simpler weight function: …See Ismail (2009, (4.10.9)) for orthogonality on the unit circle for general values of $a$. … ►In this limit the finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty })$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty })$ (see (18.34.5_5)). …