integer degree and order

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11: 31.8 Solutions via Quadratures

… ►For half-odd-integer values of the exponent parameters: … ►Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. The degree $g$ is given by … ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ may become a rational function in $z$. …The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …

12: 14.24 Analytic Continuation

… ►

14.24.1 $$P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={\mathrm{e}}^{s\nu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right)+\frac{2\mathrm{i}\sin \left(\left(\nu +\frac{1}{2}\right)s\pi \right){\mathrm{e}}^{-s\pi \mathrm{i}/2}}{\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\mu -\nu \right)}{𝑸}_{\nu }^{\mu }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

►

14.24.2 $${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={(-1)}^s{\mathrm{e}}^{-s\nu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right),$$

ⓘ

Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.23, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

… ►

14.24.3 $$P_{\nu ,s}^{-\mu }\left(z\right)={\mathrm{e}}^{s\mu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right),$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Permalink:

http://dlmf.nist.gov/14.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

►

14.24.4 $${𝑸}_{\nu ,s}^{\mu }\left(z\right)={\mathrm{e}}^{-s\mu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right)-\frac{\pi \mathrm{i}\sin \left(s\mu \pi \right)}{\sin \left(\mu \pi \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{P}_{\nu }^{-\mu }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

…

13: 14.18 Sums

… ►

14.18.9 $${𝖯}_{\nu }^{-\mu }\left(x\right)=\frac{\sin \left(\nu \pi \right)}{\pi }\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n\frac{2n+1}{(\nu -n)(\nu +n+1)}{𝖯}_n^{-\mu }\left(x\right),$$ , $\mu \ge 0$, $\nu \notin \mathbb{Z}$.

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathbb{Z}$: set of all integers, $\notin$: not an element of, $\sin z$: sine function, $n$: nonnegative integer, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.18(iii)

Permalink:

http://dlmf.nist.gov/14.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14

…

14: 14.1 Special Notation

… ► ►►►

$x$, $y$, $\tau$	real variables.
…
$m$, $n$	unless stated otherwise, nonnegative integers, used for order and degree, respectively.
…

…

15: 18.2 General Orthogonal Polynomials

… ►If the polynomials $p_n(x)$ ($n=0,1,\mathrm{\dots },N$) are orthogonal on a finite set $X$ of $N+1$ distinct points as in (18.2.3), then the polynomial $p_{N+1}(x)$ of degree $N+1$, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on $X$. … ►The Hankel determinant ${\mathrm{\Delta }}_n$ of order $n$ is defined by ${\mathrm{\Delta }}_0=1$ and … ►

Degree lowering and raising differentiation formulas and structure relations

… ►If $A_n(x)$ and $B_n(x)$ are polynomials of degree independent of $n$, and moreover ${\pi }_n(x)$ is a polynomial $\pi (x)$ independent of $n$ then … ►Polynomials $p_n(x)$ of degree $n$ ($n=0,1,2,\mathrm{\dots }$) are called Sheffer polynomials if they are generated by a generating function of the form …

16: 14.9 Connection Formulas

… ►

14.9.6 $$\pi \cos \left(\nu \pi \right)\cos \left(\mu \pi \right){𝖯}_{\nu }^{\mu }\left(x\right)=\sin \left((\nu +\mu )\pi \right){𝖰}_{\nu }^{\mu }\left(x\right)-\sin \left((\nu -\mu )\pi \right){𝖰}_{-\nu -1}^{\mu }\left(x\right).$$

ⓘ

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, $\cos z$: cosine function, $\sin z$: sine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(i), §14.9 and Ch.14

… ►

14.9.8 $$\frac{1}{2}\pi \sin \left((\nu -\mu )\pi \right){𝖯}_{\nu }^{-\mu }\left(x\right)=-\cos \left((\nu -\mu )\pi \right){𝖰}_{\nu }^{-\mu }\left(x\right)-{𝖰}_{\nu }^{-\mu }\left(-x\right),$$

ⓘ

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, $\cos z$: cosine function, $\sin z$: sine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.7(iii), §14.8(i)

Permalink:

http://dlmf.nist.gov/14.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(ii), §14.9 and Ch.14

… ►

14.9.14 $${𝑸}_{\nu }^{-\mu }\left(x\right)={𝑸}_{\nu }^{\mu }\left(x\right),$$

ⓘ

Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.6 (modified)

Referenced by:

§14.23, §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

… ►

14.9.16 $${𝑸}_{\nu }^{\mu }\left(x\right)={\left(\frac{1}{2}\pi \right)}^{1/2}{\left(x^2-1\right)}^{-1/4}{P}_{-\mu -(1/2)}^{-\nu -(1/2)}\left(x{\left(x^2-1\right)}^{-1/2}\right).$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.7 (modified)

Referenced by:

§14.19(v), §14.21(iii), §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iv), §14.9 and Ch.14

… ►

14.9.17 $$P_{\nu }^{\mu }\left(x\right)={(2/\pi )}^{1/2}{\left(x^2-1\right)}^{-1/4}{𝑸}_{-\mu -(1/2)}^{\nu +(1/2)}\left(x{\left(x^2-1\right)}^{-1/2}\right).$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.19(v), §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iv), §14.9 and Ch.14

17: 1.2 Elementary Algebra

… ►Let ${\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n$ be distinct constants, and $f(x)$ be a polynomial of degree less than $n$. … ►If $m_1,m_2,\mathrm{\dots },m_n$ are positive integers and , then there exist polynomials $f_j(x)$, , such that … ►Square $n\times n$ matrices (said to be of order $n$ ) dominate the use of matrices in the DLMF, and they have many special properties. Unless otherwise indicated, matrices are assumed square, of order $n$; and, when vectors are combined with them, these are of length $n$. … ►The norm of an order $n$ square matrix, $𝐀$, is …

18: 14.3 Definitions and Hypergeometric Representations

… ►When $\mu =m$ ($\in \mathbb{Z}$) (14.3.2) is replaced by its limiting value; see Hobson (1931, §132) for details. … ►

14.3.6

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.21(i), §14.21(iii), §14.3(ii), §14.3(iv), §15.9(iv)

Permalink:

http://dlmf.nist.gov/14.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

… ►

14.3.9

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

Permalink:

http://dlmf.nist.gov/14.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

… ►

14.3.11

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mu$: general order, $\nu$: general degree, : real variable and $w_2$: function $w_1$,

A&S Ref:

8.1.4 (modified)

Referenced by:

§10.59, §14.23, §14.3(iii), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

… ►

14.3.15

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.3.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

…

19: 14.8 Behavior at Singularities

… ►

14.8.1 $${𝖯}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.30(ii), §14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

… ►

14.8.4 $${𝖰}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{2}\cos \left(\mu \pi \right)\mathrm{\Gamma }\left(\mu \right){\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

… ►

14.8.7 $$P_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii), §14.24, §14.8(ii)

Permalink:

http://dlmf.nist.gov/14.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

… ►

14.8.11 $${𝑸}_{\nu }^{\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(\mu \right)}{2\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$ $\mathrm{\Re }\mu >0$, $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

… ►

14.8.15 $${𝑸}_{\nu }^{\mu }\left(x\right)\sim \frac{{\pi }^{1/2}}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right){(2x)}^{\nu +1}},$$ $\nu \ne -\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.8(iii)

Permalink:

http://dlmf.nist.gov/14.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(iii), §14.8 and Ch.14

…

20: 18.30 Associated OP’s

… ►The constant $c$ is usually taken as a positive integer. However, if the recurrence coefficients are polynomial, or rational, functions of $n$, polynomials of degree $n$ may be well defined for $c\in ℝ$ provided that $A_{n+c}{B}_{n+c}\ne 0,n=0,1,\mathrm{\dots }$ Askey and Wimp (1984). ►The order $c$ recurrence is initialized as …The lowest order monic versions of both of these appear in §18.2(x), (18.2.31) defining the $c=1$ associated monic polynomials, and (18.2.32) their closely related cousins the $c=0$ corecursive polynomials. … ►The zeroth order corecursive monic polynomials ${\widehat{p}}_n^{(0)}(x)$ follow directly from the alternate initialization …