integrals with respect to degree

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1: 14.20 Conical (or Mehler) Functions

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§14.20(x) Zeros and Integrals

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2: 37.2 General Orthogonal Polynomials of Two Variables

… ►The space ${𝒱}_n$ of orthogonal polynomials of degree $n$ consists of all $P\in {\mathrm{\Pi }}_n$ such that ${⟨P,Q⟩}_W=0$ for all $Q\in {\mathrm{\Pi }}_{n-1}$ ($n>0$, otherwise ${𝒱}_0={\mathrm{\Pi }}_0$). …

3: 7.23 Tables

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

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4: Bibliography C

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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.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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5: 18.33 Polynomials Orthogonal on the Unit Circle

… ►A system of polynomials $\{{\varphi }_n(z)\}$, $n=0,1,\mathrm{\dots }$, where ${\varphi }_n(z)$ is of proper degree $n$, is orthonormal on the unit circle with respect to the weight function $w(z)$ ($\ge 0$) if … ►Let $\{p_n(x)\}$ and $\{q_n(x)\}$, $n=0,1,\mathrm{\dots }$, 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 $\{{\mathrm{\Phi }}_n(z)\}$, $n=0,1,\mathrm{\dots }$, where ${\mathrm{\Phi }}_n(x)$ 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 $p^{\ast }(z)$ by …

6: 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, $s\ge 2$). … ►and the coefficients ${𝖠}_s(\tau )$ 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 $\mu \to \mathrm{\infty }$ uniformly with respect to $t\in [1+\delta ,\mathrm{\infty })$. … ►where $\xi ,\eta$ are given by (12.10.7), (12.10.23), respectively, and …

7: 18.2 General Orthogonal Polynomials

… ►A system (or set) of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, 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 $w(x)$ ($\ge 0$) if … ►For OP’s $\{p_n(x)\}$ on $ℝ$ 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 $w(x)$ on $(-1,1)$. … ►The monic OP’s $p_n(x)$ with respect to the measure $d\mu (x)$ can be expressed in terms of the moments by … ►

Degree lowering and raising differentiation formulas and structure relations

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8: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

§8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(i) Large $z$

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). … ►

§8.20(ii) Large $p$

… ►so that $A_k(\lambda )$ is a polynomial in $\lambda$ of degree $k-1$ when $k\ge 1$. …

9: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►Define an inner product …Let ${𝒱}_n^d$ denote the space of OPs of degree $n$ of $d$ variables, i. … ►It is natural to use a graded order by ordering monomials by the total degree and choose a linear order (lexicographical order, for example) for monomials of the same total degree. …In the co-monic basis ${\{Q_{𝝂}\}}_{|𝝂|=n}$, biorthogonal to the monic basis, $Q_{𝝂}$ is a polynomial of degree $n$ which is orthogonal to ${𝐱}^{𝝁}$ ($|𝝁|\le n$, $𝝁\ne 𝝂$) with respect to the inner product (37.13.1), analogous to (37.2.4). … ►, having discrete OPs of degree $n$ as eigenspaces, have been completely classified by Iliev and Xu (2007). …

10: 37.17 Hermite Polynomials on ${ℝ}^d$

… ►The OPs of degree $n$ with respect to the inner product (37.17.1) form the space ${𝒱}_n({ℝ}^d)$. … ►Specialization in §37.13(i) of the rotation invariant weight function to $W(𝐱)=\exp \left(-{\Vert 𝐱\Vert }^2\right)$ gives for the corresponding OPs that … ►

37.17.10 $$\left(\frac{1}{2}\mathrm{\Delta }-\sum _{\mathrm{\ell }=1}^d{x}_{\mathrm{\ell }}{D}_{x_{\mathrm{\ell }}}\right)u(x)=-nu(x),$$ $u\in {𝒱}_n({ℝ}^d)$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $ℝ$: real line, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, ${ℝ}^d$: $d$-dimensional Euclidean space, $x$: real variable and $u$: 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 and Ch.37

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37.17.13 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{2k}{R}_{Y,k,n}^{\alpha }({\alpha }^{-\frac{1}{2}}𝐱)={(-1)}^{k}k!S_{Y,k,n}(𝐱),$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y\in {\mathscr{H}}_{n-2k}^{0,d}$.

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Symbols:

$\in$: element of, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $n$: nonnegative integer, $k$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space and ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$

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 and Ch.37

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37.17.16 $$H_{𝝂}(𝐱;𝐀)={(-1)}^{|𝝂|}{\mathrm{e}}^{⟨𝐀𝐱,𝐱⟩}{D}_{𝐱}^{𝝂}{\mathrm{e}}^{-⟨𝐀𝐱,𝐱⟩},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $⟨𝐮,𝐯⟩$: inner product over vectors and $𝐱$: 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 and Ch.37

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