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1: 28.12 Definitions and Basic Properties

… ►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. … ►

§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►For $q=0$, … ► … ►

2: 28.2 Definitions and Basic Properties

… ►

§28.2(vi) Eigenfunctions

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3: 14.11 Derivatives with Respect to Degree or Order

§14.11 Derivatives with Respect to Degree or Order

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14.11.1 $$\frac{\partial }{\partial \nu }{𝖯}_{\nu }^{\mu }\left(x\right)=\pi \cot \left(\nu \pi \right){𝖯}_{\nu }^{\mu }\left(x\right)-\frac{1}{\pi }{𝖠}_{\nu }^{\mu }(x),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.2 $$\frac{\partial }{\partial \nu }{𝖰}_{\nu }^{\mu }\left(x\right)=-\frac{1}{2}{\pi }^2{𝖯}_{\nu }^{\mu }\left(x\right)+\frac{\pi \sin \left(\mu \pi \right)}{\sin \left(\nu \pi \right)\sin \left((\nu +\mu )\pi \right)}{𝖰}_{\nu }^{\mu }\left(x\right)-\frac{1}{2}\cot \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(x)+\frac{1}{2}\csc \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(-x),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

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Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

…

4: Bibliography S

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A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

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Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.

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External Links:

Document

Referenced by:

§22.19(ii).

Encodings:

BibTeX

… ►

On the derivative of the Legendre function of the first kind with respect to its degree. J. Phys. A 39 (49), pp. 15147–15172.

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External Links:

ISSN 0305-4470, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

…

5: 14.20 Conical (or Mehler) Functions

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14.20.19 $$\alpha =\mu /\tau ,$$

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Symbols:

$\tau$: real variable, $\mu$: general order and $\alpha$

Permalink:

http://dlmf.nist.gov/14.20.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(viii), §14.20 and Ch.14

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§14.20(ix) Asymptotic Approximations: Large $\mu$, $0\le \tau \le A\mu$

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14.20.23 $$\beta =\tau /\mu ,$$

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Symbols:

$\tau$: real variable, $\mu$: general order and $\beta$

Permalink:

http://dlmf.nist.gov/14.20.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(ix), §14.20 and Ch.14

… ►For the case of purely imaginary order and argument see Dunster (2013). ►

§14.20(x) Zeros and Integrals

…

6: 37.18 Orthogonal Polynomials on Quadratic Domains

… ►Let ${𝒱}_n({𝕍}^{d+1},W)$ be the space of orthogonal polynomials of degree $n$ with respect to the inner product. … ►For a fixed $m\in {\mathbb{N}}_0$, let $q_n^{(m)}$ be the orthogonal polynomial of degree $n$ with respect to the weight function ${|\varphi (t)|}^{2m+d}{w}_1(t)$ on $(a,b)$ and let $\{P_{𝐤}^m:|𝐤|=n,𝐤\in {\mathbb{N}}_0^d\}$ be an orthonormal basis for ${𝒱}_n({𝔹}^d,w_2)$, the space of orthogonal polynomials of degree $n$ with respect to $w_2(\Vert 𝐱\Vert )$ on the unit ball ${𝔹}^d$. … ►Only for $\beta =0$ the spaces ${𝒱}_n({𝕍}^{d+1},W_{\mu ,\beta ,\gamma })$ are eigenspaces of a second order partial differential operator: … ►The spaces ${𝒱}_n({𝕍}^{d+1},W_{\mu ,0})$ are eigenspaces of a second order partial differential operator: …

7: 1.1 Special Notation

… ► ►►►►►

$x,y$	real variables.
…
$L^2(X,d\alpha )$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $d\alpha$.
…
$\mathrm{deg}$	degree.
primes	derivatives with respect to the variable, except where indicated otherwise.
…

►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

8: Bibliography C

… ►

An algorithm for exponential integrals of real order. Computing 45 (3), pp. 269–276.

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Notes:

Includes Fortran code

External Links:

ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§8.25(v), §8.28(vi).

Encodings:

BibTeX

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The zeros of Hankel functions as functions of their order. Numer. Math. 7 (3), pp. 238–250.

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External Links:

ISSN 0029-599X, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

BibTeX

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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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Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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Zeros of the Hankel function of real order and of its derivative. Math. Comp. 39 (160), pp. 639–645.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Ahmed), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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9: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ►Let ${𝒱}_n({𝕍}_0^{d+1})$ be the space of orthogonal polynomials of degree $n$ with respect to the inner product. … ►For a fixed $m\in {\mathbb{N}}_0$, let $q_n^{(m)}$ be an orthogonal polynomial of degree $n$ with respect to the weight function $\varphi {(t)}^{2m+d-1}w(t)$ on $(a,b)$. … ►The spaces ${𝒱}_n({𝕍}_0^{d+1},w_{-1,\gamma })$ are eigenspaces of a second order partial differential operator: … ►Only for $\beta =-1$ the spaces ${𝒱}_n({𝕍}_0^{d+1},w_{\beta })$ are eigenspaces of a second order partial differential operator: …

10: 37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

… ►See (Koornwinder, 1974b, §5) for a fourth order differential operator which also has the polynomials $p_{k,n}^{\alpha ,\beta ,\gamma }$ as eigenfunctions, and such that its eigenvalue together with the eigenvalue (37.8.7) completely determine $k,n$. … ►are OPs of degree $n$ with respect to the weight function …