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1: 14.14 Continued Fractions

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14.14.1 $$\frac{1}{2}{\left(x^2-1\right)}^{1/2}\frac{P_{\nu }^{\mu }\left(x\right)}{P_{\nu }^{\mu -1}\left(x\right)}=\frac{x_0}{y_0+}\frac{x_1}{y_1+}\frac{x_2}{y_2+}\mathrm{\cdots },$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

►where … ►

14.14.3 $$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$ $\nu \ne \mu$,

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

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2: 18.13 Continued Fractions

… ► $T_n\left(x\right)$ is the denominator of the $n$th approximant to: …and $U_n\left(x\right)$ is the denominator of the $n$th approximant to: … ► $P_n\left(x\right)$ is the denominator of the $n$th approximant to: … ► $L_n\left(x\right)$ is the denominator of the $n$th approximant to: … ► $H_n\left(x\right)$ is the denominator of the $n$th approximant to: …

3: 1.12 Continued Fractions

… ► $A_n$ and $B_n$ are called the $n$th (canonical) numerator and denominator respectively. … ►

1.12.7 $$A_n{B}_{n-1}-B_n{A}_{n-1}={(-1)}^{n-1}\prod _{k=1}^n{a}_k,$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$k$: integer, $n$: nonnegative integer, $A_n$: $n$th numerator and $B_n$: $n$th denominator

Permalink:

http://dlmf.nist.gov/1.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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1.12.11 $$a_n=\frac{B_n}{B_{n-2}}\frac{C_{n-1}-C_n}{C_{n-1}-C_{n-2}},$$ $n=2,3,4,\mathrm{\dots }$,

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Symbols:

$n$: nonnegative integer, $B_n$: $n$th denominator and $C_n(w)$: continued fraction

Permalink:

http://dlmf.nist.gov/1.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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1.12.13 $$b_n=\frac{B_n}{B_{n-1}}\frac{C_n-C_{n-2}}{C_{n-1}-C_{n-2}},$$ $n=2,3,4,\mathrm{\dots }$,

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Symbols:

$n$: nonnegative integer, $B_n$: $n$th denominator and $C_n(w)$: continued fraction

Permalink:

http://dlmf.nist.gov/1.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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4: 18.30 Associated OP’s

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Numerator and Denominator Polynomials

►The $p_n^{(0)}(x)$ are also referred to as the numerator polynomials, the $p_n(x)$ then being the denominator polynomials, in that the $n$-th approximant of the continued fraction, $z\in ℂ$, … ►

Markov’s Theorem

►The ratio $p_n^{(0)}(z)/p_n(z)$, as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. …

5: 33.8 Continued Fractions

… ►The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision. …

6: 3.10 Continued Fractions

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3.10.2 $$C_n=b_0+\frac{a_1}{b_1+}\frac{a_2}{b_2+}\mathrm{\cdots }\frac{a_n}{b_n}=\frac{A_n}{B_n}.$$

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Defines:

$A_n$: continued fraction numerator (locally), $B_n$: continued fraction denominator (locally) and $C_n$: continued fraction approximant (locally)

Referenced by:

§3.10(iii), §3.10(iii)

Permalink:

http://dlmf.nist.gov/3.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.10(i), §3.10 and Ch.3

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3.10.4 $$\frac{A_n}{B_n}=u_0+u_1+\mathrm{\cdots }+u_n,$$ $n=0,1,\mathrm{\dots }$.

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Symbols:

$A_n$: continued fraction numerator, $B_n$: continued fraction denominator and $u_n$: series

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/3.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §3.10(ii), §3.10 and Ch.3

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7: 23.12 Asymptotic Approximations

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23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

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8: 24.10 Arithmetic Properties

… ►The denominator of $B_{2n}$ is the product of all these primes $p$. …

9: 33.6 Power-Series Expansions in $\rho$

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33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

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10: 17.11 Transformations of $q$-Appell Functions

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17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.11.2)

Permalink:

http://dlmf.nist.gov/17.11.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

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