Olver’s

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1: 32.17 Methods of Computation

… ►For numerical studies of ${\text{P}}_{\text{II}}$ see Rosales (1978), Miles (1978, 1980), Kashevarov (1998, 2004), and S.  Olver (2011). …

2: 15.1 Special Notation

… ►and also ►

15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ 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 and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

…

3: 14.21 Definitions and Basic Properties

… ►

14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.21(i), §14.21 and Ch.14

…

4: Frank W. J. Olver

… ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … ►In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olver’s 65th birthday, with Proceedings published by Marcel Dekker in 1990. … ►Fitting capstones to Olver’s long career, these works are on track to extend the legacy of the classic Abramowitz and Stegun handbook well into the 21st century. According to DLMF Project Lead Daniel Lozier, “Olver’s encyclopedic knowledge of the field, his clear vision for mathematical exposition, his keen sense of the needs of practitioners, and his unfailing attention to detail were key to the success of that project. …

5: 13.1 Special Notation

… ►The main functions treated in this chapter are the Kummer functions $M(a,b,z)$ and $U(a,b,z)$, Olver’s function $𝐌(a,b,z)$, and the Whittaker functions $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$. …

6: 14.3 Definitions and Hypergeometric Representations

… ►

14.3.3

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and : real variable

Permalink:

http://dlmf.nist.gov/14.3.E3

Encodings:

pMML, png, TeX

See also:

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

►is Olver’s hypergeometric function (§15.1). … ►

14.3.6

ⓘ

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

… ►

14.3.10

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second 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, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.20(i), §14.21(i), §14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.3.E10

Encodings:

pMML, png, TeX

See also:

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

… ►

14.3.19

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $𝐅(a,b;c;z)$ 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.E19

Encodings:

pMML, png, TeX

See also:

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

…

7: 13.10 Integrals

… ►

13.10.1 $$\int 𝐌(a,b,z)dz=\frac{1}{a-1}𝐌(a-1,b-1,z),$$

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(i), §13.10 and Ch.13

… ►

13.10.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;k/z),$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

►

13.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,b,t)dt=z^{-b}{\left(1-\frac{1}{z}\right)}^{-a},$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>1$,

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

►

13.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{b-1}𝐌(a,c,t)dt=\frac{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)},$$ $\mathrm{\Re }\left(c-a\right)>\mathrm{\Re }b>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Permalink:

http://dlmf.nist.gov/13.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

… ►

13.10.10 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}𝐌(a,b,-t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-\lambda \right)},$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iii), §13.10 and Ch.13

…

8: 15.7 Continued Fractions

… ►

15.7.1 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=t_0-\frac{u_{1}z}{t_1-}\frac{u_{2}z}{t_2-}\frac{u_{3}z}{t_3-}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ 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, $t_n$: coefficient and $u_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

… ►

15.7.3 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=v_0-\frac{w_1}{v_1-}\frac{w_2}{v_2-}\frac{w_3}{v_3-}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ 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, $v_n$: coefficient and $w_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

… ►

15.7.5 $$\frac{𝐅(a,b;c;z)}{𝐅(a+1,b+1;c+1;z)}=x_0+\frac{y_1}{x_1+}\frac{y_2}{x_2+}\frac{y_3}{x_3+}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $y$: real variable, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

…

9: 15.15 Sums

… ►

15.15.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};\frac{1}{z})={\left(1-\frac{z_0}{z}\right)}^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{s!}𝐅(\genfrac{}{}{0pt}{}{-s,b}{c};\frac{1}{z_0}){\left(1-\frac{z}{z_0}\right)}^{-s}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.19(i)

Permalink:

http://dlmf.nist.gov/15.15.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §15.15 and Ch.15

…

10: 3.6 Linear Difference Equations

… ►

§3.6(v) Olver’s Algorithm

… ►The backward recursion can be carried out using independently computed values of $J_N\left(1\right)$ and $J_{N+1}\left(1\right)$ or by use of Miller’s algorithm (§3.6(iii)) or Olver’s algorithm (§3.6(v)). … ►Thus the asymptotic behavior of the particular solution ${𝐄}_n\left(1\right)$ is intermediate to those of the complementary functions $J_n\left(1\right)$ and $Y_n\left(1\right)$; moreover, the conditions for Olver’s algorithm are satisfied. … ►

Table 3.6.1: Weber function $w_n={𝐄}_n\left(1\right)$ computed by Olver’s algorithm.

► ►►

$n$	$p_n$		$e_n$		$e_n/(p_n{p}_{n+1})$		$w_n$
…

►

…