big q-Jacobi polynomials

SearchAdvancedHelp

(0.003 seconds)

11—20 of 314 matching pages

11: 18.35 Pollaczek Polynomials

§18.35 Pollaczek Polynomials

… ►There are 3 types of Pollaczek polynomials: … ►For the monic polynomials … ► … ► …

12: 24.17 Mathematical Applications

§24.17 Mathematical Applications

… ►Let ${𝒮}_n$ denote the class of functions that have $n-1$ continuous derivatives on $ℝ$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in \mathbb{Z}$. … ►

§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)).

13: 2.10 Sums and Sequences

… ►As in §24.2, let $B_n$ and $B_n\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and ${\tilde{B}}_n\left(x\right)$ the $n$th Bernoulli periodic function $B_n\left(x-\lfloor x\rfloor \right)$. … ►From §24.12(i), (24.2.2), and (24.4.27), ${\tilde{B}}_{2m}\left(x\right)-B_{2m}$ is of constant sign ${(-1)}^m$. … ►

On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

2.10.30 $$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$ $z\to z_j$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

where ${\sigma }_j$ is a positive constant.

… ►

Example

►Let $\alpha$ be a constant in $(0,2\pi )$ and $P_n$ denote the Legendre polynomial of degree $n$. …

14: 10.41 Asymptotic Expansions for Large Order

… ►Also, $U_k(p)$ and $V_k(p)$ are polynomials in $p$ of degree $3k$, given by $U_0(p)=V_0(p)=1$, and … ►

10.41.12 $$I_{\nu }\left(\nu z\right)=\frac{{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}{(1+z^2)}^{\frac{1}{4}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{U_k(p)}{{\nu }^k}+O\left(\frac{1}{z^{\mathrm{\ell }}}\right)\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\eta$ , $p$ and $U_k(p)$: polynomial coefficient

Referenced by:

§10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(iv), §10.41 and Ch.10

►

10.41.13 $$K_{\nu }\left(\nu z\right)={\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}{(-1)}^k\frac{U_k(p)}{{\nu }^k}+O\left(\frac{1}{z^{\mathrm{\ell }}}\right)\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\eta$ , $p$ and $U_k(p)$: polynomial coefficient

Referenced by:

§10.41(iv), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(iv), §10.41 and Ch.10

… ►

10.41.14 $$J_{\nu }\left(\nu z\right)={\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }}\frac{A_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+3}}\right)\right)+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{B_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+1}}\right)\right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

Referenced by:

§10.41(v)

Permalink:

http://dlmf.nist.gov/10.41.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(v), §10.41 and Ch.10

►

10.41.15 $$Y_{\nu }\left(\nu z\right)=-{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }}\frac{A_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+3}}\right)\right)+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{B_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+1}}\right)\right)\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

Referenced by:

§10.41(v)

Permalink:

http://dlmf.nist.gov/10.41.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(v), §10.41 and Ch.10

…

15: 13.2 Definitions and Basic Properties

… ►When $a=-n$, $n=0,1,2,\mathrm{\dots }$, $𝐌(a,b,z)$ is a polynomial in $z$ of degree not exceeding $n$; this is also true of $M(a,b,z)$ provided that $b$ is not a nonpositive integer. … ►When $a=-m$, $m=0,1,2,\mathrm{\dots }$, $U(a,b,z)$ is a polynomial in $z$ of degree $m$: … ►

13.2.13 $$M(a,b,z)=1+O\left(z\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

… ►Except when $a=0,-1,\mathrm{\dots }$ (polynomial cases), …

16: 22.10 Maclaurin Series

… ►

22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.2 $$\mathrm{cn}(z,k)=1-\frac{z^2}{2!}+\left(1+4k^2\right)\frac{z^4}{4!}-\left(1+44k^2+16k^4\right)\frac{z^6}{6!}+O\left(z^8\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.2

Permalink:

http://dlmf.nist.gov/22.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.3 $$\mathrm{dn}(z,k)=1-k^2\frac{z^2}{2!}+k^2\left(4+k^2\right)\frac{z^4}{4!}-k^2\left(16+44k^2+k^4\right)\frac{z^6}{6!}+O\left(z^8\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.3

Permalink:

http://dlmf.nist.gov/22.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

… ►

22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

… ►

22.10.6 $$\mathrm{dn}(z,k)=1-\frac{k^2}{2}{\sin }^{2}z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.3

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

…

17: 15.12 Asymptotic Approximations

… ►

15.12.2 $$F(a,b;c;z)=\sum _{s=0}^{m-1}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s+O\left(c^{-m}\right),$$ $|c|\to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

Referenced by:

§15.19(iv)

Permalink:

http://dlmf.nist.gov/15.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(ii), §15.12 and Ch.15

… ►

15.12.5 $$𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b-\lambda }{c};\frac{1}{2}-\frac{1}{2}z)=2^{(a+b-1)/2}\frac{{(z+1)}^{(c-a-b-1)/2}}{{(z-1)}^{c/2}}\sqrt{\zeta \sinh \zeta }{\left(\lambda +\frac{1}{2}a-\frac{1}{2}b\right)}^{1-c}\left(I_{c-1}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)(1+O({\lambda }^{-2}))+\frac{I_{c-2}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)}{2\lambda +a-b}\left(\left(c-\frac{1}{2}\right)\left(c-\frac{3}{2}\right)\left(\frac{1}{\zeta }-\coth \zeta \right)+\frac{1}{2}(2c-a-b-1)(a+b-1)\tanh \left(\frac{1}{2}\zeta \right)+O({\lambda }^{-2})\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $I_{\nu }\left(z\right)$: modified Bessel 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, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1). … ►

15.12.7 $$F(\genfrac{}{}{0pt}{}{a,b-\lambda }{c+\lambda };-z)=2^{b-c+(1/2)}{\left(\frac{z+1}{2\sqrt{z}}\right)}^{\lambda }\left({\lambda }^{a/2}U(a-\frac{1}{2},-\alpha \sqrt{\lambda })\left({(1+z)}^{c-a-b}{z}^{1-c}{\left(\frac{\alpha }{z-1}\right)}^{1-a}+O({\lambda }^{-1})\right)+\frac{{\lambda }^{(a-1)/2}}{\alpha }U(a-\frac{3}{2},-\alpha \sqrt{\lambda })\left({(1+z)}^{c-a-b}{z}^{1-c}{\left(\frac{\alpha }{z-1}\right)}^{1-a}-2^{c-b-(1/2)}{\left(\frac{\alpha }{z-1}\right)}^a+O({\lambda }^{-1})\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\alpha$

Permalink:

http://dlmf.nist.gov/15.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.9 $${(z+1)}^{3\lambda /2}{(2\lambda )}^{c-1}𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b+2\lambda }{c};-z)={\lambda }^{-1/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_0(\zeta )+O({\lambda }^{-1})\right)+{\lambda }^{-2/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_1(\zeta )+O({\lambda }^{-1})\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $a_j(\zeta )$: function

Referenced by:

§15.12(iii)

Permalink:

http://dlmf.nist.gov/15.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

…

18: 14.15 Uniform Asymptotic Approximations

… ►

14.15.1 $${𝖯}_{\nu }^{-\mu }\left(\pm x\right)={\left(\frac{1\mp x}{1\pm x}\right)}^{\mu /2}\left(\sum _{j=0}^{J-1}\frac{{\left(\nu +1\right)}_j{\left(-\nu \right)}_j}{j!\mathrm{\Gamma }\left(j+1+\mu \right)}{\left(\frac{1\mp x}{2}\right)}^j+O\left(\frac{1}{\mathrm{\Gamma }\left(J+1+\mu \right)}\right)\right)$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $x$: real variable, $\mu$: general order, $\nu$: general degree and $J$: nonnegative integer

Referenced by:

§14.15(i)

Permalink:

http://dlmf.nist.gov/14.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(i), §14.15 and Ch.14

… ►

14.15.3 $${𝑸}_{\nu }^{\mu }\left(x\right)=\frac{1}{{\mu }^{\nu +(1/2)}}{\left(\frac{\pi u}{2}\right)}^{1/2}{I}_{\nu +\frac{1}{2}}\left(\mu u\right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree and $u$

Referenced by:

§14.15(i)

Permalink:

http://dlmf.nist.gov/14.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(i), §14.15 and Ch.14

… ►

14.15.13 $$P_{\nu }^{-\mu }\left(\cosh \xi \right)=\frac{1}{{\nu }^{\mu }}{\left(\frac{\xi }{\sinh \xi }\right)}^{1/2}{I}_{\mu }\left(\left(\nu +\frac{1}{2}\right)\xi \right)\left(1+O\left(\frac{1}{\nu }\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mu$: general order, $\nu$: general degree and $\xi$: variable

Permalink:

http://dlmf.nist.gov/14.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iii), §14.15 and Ch.14

… ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_n\left(\cos \theta \right)$ as $n\to \mathrm{\infty }$ with $\theta$ fixed. … ►

14.15.17 $$P_{\nu }^{-\mu }\left(x\right)=\beta {\left(\frac{{\alpha }^2-y}{x^2-1+{\alpha }^2}\right)}^{1/4}{I}_{\mu }\left(\left(\nu +\frac{1}{2}\right){|y|}^{1/2}\right)\left(1+O\left(\frac{1}{\nu }\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\beta$ , $y$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iv), §14.15 and Ch.14

…

19: 3.5 Quadrature

… ►

Gauss–Legendre Formula

… ►The $p_n(x)$ are the monic Hermite polynomials $H_n\left(x\right)$ (§18.3). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … … ►We choose $s=1$ so that $f(\zeta )=O\left(1\right)$ at infinity. …

20: 29.7 Asymptotic Expansions

… ►

29.7.5 $$b_{\nu }^{m+1}\left(k^2\right)-a_{\nu }^m\left(k^2\right)=O\left({\nu }^{m+\frac{3}{2}}{\left(\frac{1-k}{1+k}\right)}^{\nu }\right),$$ $\nu \to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Referenced by:

§29.7(i)

Permalink:

http://dlmf.nist.gov/29.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

… ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu \to \mathrm{\infty }$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …