integrals with respect to degree

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1: 14.20 Conical (or Mehler) Functions

… ►

§14.20(x) Zeros and Integrals

…

2: 37.2 General Orthogonal Polynomials of Two Variables

… ►The space

\mathcal{V}_{n}

of orthogonal polynomials of degree $n$ consists of all

P\in\Pi_{n}

such that

\left\langle P,Q\right\rangle_{W}=0

for all

Q\in\Pi_{n-1}

(

n>0

, otherwise

\mathcal{V}_{0}=\Pi_{0}

). …

Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

…

5: Bibliography C

… ►

H. S. Cohl (2010) Derivatives with respect to the degree and order of associated Legendre functions for

|z|>1

using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

ⓘ

External Links:: ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

…

6: 18.33 Polynomials Orthogonal on the Unit Circle

… ►A system of polynomials

\{\phi_{n}(z)\}

n=0,1,\dots

, where

\phi_{n}(z)

is of proper degree

n

, is orthonormal on the unit circle with respect to the weight function

w(z)

(

\geq 0

) if … ►Let

\{p_{n}(x)\}

and

\{q_{n}(x)\}

n=0,1,\dots

, be OP’s with weight functions

w_{1}(x)

and

w_{2}(x)

, respectively, on

(-1,1)

. … ►See Askey (1982a) and Pastro (1985) for special cases extending (18.33.13)–(18.33.14) and (18.33.15)–(18.33.16), respectively. … ►A system of monic polynomials

\{\Phi_{n}(z)\}

n=0,1,\dots

, where

\Phi_{n}(x)

is of proper degree

n

, is orthogonal on the unit circle with respect to the measure

\mu

if … ►with complex coefficients

c_{k}

and of a certain degree

n

define the reversed polynomial

p^{*}(z)

by …

7: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►In this section we give asymptotic expansions of PCFs for large values of the parameter

a

that are uniform with respect to the variable

z

, when both

a

and

z

(=x)

are real. … ►where

u_{s}(t)

and

v_{s}(t)

are polynomials in

t

of degree

3s

, (

s

odd),

3s-2

(

s

even,

s\geq 2

). … ►and the coefficients

\mathsf{A}_{s}(\tau)

are the product of

\tau^{s}

and a polynomial in

\tau

of degree

2s

. … ►The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when

\mu\to\infty

uniformly with respect to

t\in[1+\delta,\infty)

. … ►where

\xi,\eta

are given by (12.10.7), (12.10.23), respectively, and …

8: 18.2 General Orthogonal Polynomials

… ►A system (or set) of polynomials

\{p_{n}(x)\}

n=0,1,2,\ldots

, where

p_{n}(x)

has degree

n

as in §18.1(i), is said to be orthogonal on

(a,b)

with respect to the weight function

w(x)

(

\geq 0

) if … ►For OP’s

\{p_{n}(x)\}

\mathbb{R}

with respect to an even weight function

w(x)

we have … ►As a slight variant let

\{p_{n}(x)\}

be OP’s with respect to an even weight function

w(x)

(-1,1)

. … ►The monic OP’s

p_{n}(x)

with respect to the measure

\,\mathrm{d}\mu(x)

can be expressed in terms of the moments by … ►

Degree lowering and raising differentiation formulas and structure relations

…

9: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

… ►The OPs of degree

n

with respect to the inner product (37.17.1) form the space

\mathcal{V}_{n}({\mathbb{R}}^{d})

. … ►Specialization in §37.13(i) of the rotation invariant weight function to

W(\mathbf{x})=\exp\left(-{\left\|{\mathbf{x}}\right\|}^{2}\right)

gives for the corresponding OPs that … ►

37.17.10

\left(\tfrac{1}{2}\Delta-\sum_{\ell=1}^{d}x_{\ell}{D}_{x_{\ell}}\right)u(x)=-n% \,u(x),

u\in\mathcal{V}_{n}({\mathbb{R}}^{d})

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\in$ : element of, $\mathbb{R}$ : real line, $n$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , ${\mathbb{R}}^{d}$ : $d$ -dimensional Euclidean space, $\ell$ : positive integer and $x$ : real variable
Proved:: Dunkl and Xu (2014, (5.1.7))(proved)
Referenced by:: §37.13(ii), §37.17
Permalink:: http://dlmf.nist.gov/37.17.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §37.17(iii), §37.17, Part 37.III and Ch.37

… ►

37.17.13

\lim_{\alpha\to\infty}\alpha^{2k}R_{Y,k,n}^{\alpha}(\alpha^{-\frac{1}{2}}% \mathbf{x})=(-1)^{k}k!\,S_{Y,k,n}(\mathbf{x}),

0\leq k\leq\left\lfloor\frac{1}{2}n\right\rfloor

Y\in\mathcal{H}_{n-2k}^{0,d}

ⓘ

Symbols:: $\in$ : element of, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : nonnegative integer, $n$ : nonnegative integer, $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space and $\mathcal{H}_{n}^{0,d}$ : space of spherical harmonics of degree $n$ on $\mathbb{S}^{d-1}$
Proof sketch:: Use (37.15.7) and (18.7.22).
Permalink:: http://dlmf.nist.gov/37.17.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §37.17(v), §37.17, Part 37.III and Ch.37

… ►

37.17.16

H_{\boldsymbol{{\nu}}}(\mathbf{x};\mathbf{A})=(-1)^{|{\boldsymbol{{\nu}}}|}{% \mathrm{e}}^{\left\langle\mathbf{A}\mathbf{x},\mathbf{x}\right\rangle}{{D}_{% \mathbf{x}}}^{\boldsymbol{{\nu}}}\,{\mathrm{e}}^{-\left\langle\mathbf{A}% \mathbf{x},\mathbf{x}\right\rangle},

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space
Source:: Erdélyi et al. (1953b, (20) in §12.8)
Permalink:: http://dlmf.nist.gov/37.17.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §37.17(vi), §37.17(vi), §37.17, Part 37.III and Ch.37

…

10: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^{2}-y^{2}}$

… ►The OPs of degree

n

with respect to the inner product (37.6.1) form the space

\mathcal{V}_{n}

. … ►There is an obvious orthogonal basis of

\mathcal{V}_{n}

consisting of products of Hermite polynomials: … ►Then the polynomials

S_{m,n-m}(x+\mathrm{i}y,x-\mathrm{i}y)

(

m=0,1,\ldots,n

) form an orthogonal basis of the space

\mathcal{V}_{n}\otimes\mathbb{C}

of complex-valued orthogonal polynomials of degree

n

{\mathbb{R}}^{2}

with weight function

{\mathrm{e}}^{-x^{2}-y^{2}}

. … ►As in (37.2.26) we can take the real and imaginary parts of (37.6.3) in order to obtain the real circular Hermite polynomials … ►The explicit basis functions in §37.4 of (bi)orthogonal polynomials on the unit disk for the weight function (37.4.2) all tend after rescaling, as

\alpha\to\infty

, to basis functions given above of OPs on

{\mathbb{R}}^{2}

for the weight function

{\mathrm{e}}^{-x^{2}-y^{2}}

: …

integrals with respect to degree

1—10 of 41 matching pages

1: 14.20 Conical (or Mehler) Functions

§14.20(x) Zeros and Integrals

2: 37.2 General Orthogonal Polynomials of Two Variables

3: null

4: 7.23 Tables

5: Bibliography C

6: 18.33 Polynomials Orthogonal on the Unit Circle

7: 12.10 Uniform Asymptotic Expansions for Large Parameter

8: 18.2 General Orthogonal Polynomials

Degree lowering and raising differentiation formulas and structure relations

9: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

10: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^{2}-y^{2}}$

integrals with respect to degree

1—10 of 41 matching pages

1: 14.20 Conical (or Mehler) Functions

§14.20(x) Zeros and Integrals

2: 37.2 General Orthogonal Polynomials of Two Variables

3: null

4: 7.23 Tables

5: Bibliography C

6: 18.33 Polynomials Orthogonal on the Unit Circle

7: 12.10 Uniform Asymptotic Expansions for Large Parameter

8: 18.2 General Orthogonal Polynomials

Degree lowering and raising differentiation formulas and structure relations

9: 37.17 Hermite Polynomials on ℝ d

10: 37.6 Plane with Weight Function e − x 2 − y 2

9: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

10: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^{2}-y^{2}}$