hypergeometric%0Afunction

SearchAdvancedHelp

(0.003 seconds)

21—30 of 754 matching pages

21: 13.30 Tables

… ►

Žurina and Osipova (1964) tabulates $M(a,b,x)$ and $U(a,b,x)$ for $b=2$, $a=-0.98(.02)1.10$, $x=0(.01)4$, 7D or 7S.

►

Slater (1960) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)10$, 7–9S; $M(a,b,1)$ for $a=-11(.2)2$ and $b=-4(.2)1$, 7D; the smallest positive $x$-zero of $M(a,b,x)$ for $a=-4(.1)-0.1$ and $b=0.1(.1)2.5$, 7D.

►

Abramowitz and Stegun (1964, Chapter 13) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)1(1)10$, 8S. Also the smallest positive $x$-zero of $M(a,b,x)$ for $a=-1(.1)-0.1$ and $b=0.1(.1)1$, 7D.

►

Zhang and Jin (1996, pp. 411–423) tabulates $M(a,b,x)$ and $U(a,b,x)$ for $a=-5(.5)5$, $b=0.5(.5)5$, and $x=0.1,1,5,10,20,30$, 8S (for $M(a,b,x)$) and 7S (for $U(a,b,x)$).

…

22: 13.6 Relations to Other Functions

… ►

Hermite Polynomials

… ►

Laguerre Polynomials

… ►

Charlier Polynomials

… ►

§13.6(vi) Generalized Hypergeometric Functions

… ►For the definition of ${}_2{}^{}F_0^{}(a,a-b+1;-;-z^{-1})$ when neither $a$ nor $a-b+1$ is a nonpositive integer see §16.5. …

23: 6.11 Relations to Other Functions

… ►

6.11.1 $$E_1\left(z\right)=\mathrm{\Gamma }(0,z).$$

ⓘ

Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

5.1.45 (special case of)

Referenced by:

§6.11

Permalink:

http://dlmf.nist.gov/6.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

►

Confluent Hypergeometric Function

►

6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

►

6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

24: 16.12 Products

§16.12 Products

►

16.12.1 $${}_0{}^{}F_1^{}(-;a;z){}_0{}^{}F_1^{}(-;b;z)={}_2{}^{}F_3^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)}{a,b,a+b-1};4z).$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

(10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8

Permalink:

http://dlmf.nist.gov/16.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

… ►

16.12.2 $${\left({}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{a+b+\frac{1}{2}};z)\right)}^2={}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{2a,2b,a+b}{a+b+\frac{1}{2},2a+2b};z).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)

Permalink:

http://dlmf.nist.gov/16.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

… ►

16.12.3 $${\left({}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};z)\right)}^2=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(2a\right)}_k{\left(2b\right)}_k{\left(c-\frac{1}{2}\right)}_k}{{\left(c\right)}_k{\left(2c-1\right)}_{k}k!}{}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1)z^k,$$ .

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.12

Permalink:

http://dlmf.nist.gov/16.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.12 and Ch.16

…

25: 13.26 Addition and Multiplication Theorems

… ►

§13.26(i) Addition Theorems for $M_{\kappa ,\mu }\left(z\right)$

►The function $M_{\kappa ,\mu }\left(x+y\right)$ has the following expansions: … ►The function $W_{\kappa ,\mu }\left(x+y\right)$ has the following expansions: … ►

§13.26(iii) Multiplication Theorems for $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$

►To obtain similar expansions for $M_{\kappa ,\mu }\left(xy\right)$ and $W_{\kappa ,\mu }\left(xy\right)$, replace $y$ in the previous two subsections by $(y-1)x$.

26: 13.2 Definitions and Basic Properties

… ►Although $M(a,b,z)$ does not exist when $b=-n$, $n=0,1,2,\mathrm{\dots }$, many formulas containing $M(a,b,z)$ continue to apply in their limiting form. … ►In general, $U(a,b,z)$ has a branch point at $z=0$. … ►Except when $z=0$ each branch of $U(a,b,z)$ is entire in $a$ and $b$. … ►if $a-n\ne 0,-1,-2,\mathrm{\dots }$, or $M(a,n+1,z)$ and …if $a=0,-1,-2,\mathrm{\dots }$, or $M(a,n+1,z)$ and …

27: 10.39 Relations to Other Functions

… ►

Confluent Hypergeometric Functions

… ►

10.39.7 $$I_{\nu }\left(z\right)=\frac{{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(2z\right)}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.47

Referenced by:

(9.6.23), (9.6.24)

Permalink:

http://dlmf.nist.gov/10.39.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

… ►For the functions $M$, $U$, $M_{0,\nu }$, and $W_{0,\nu }$ see §§13.2(i) and 13.14(i). ►

Generalized Hypergeometric Functions and Hypergeometric Function

… ►For the functions ${}_0{}^{}F_1^{}$ and $𝐅$ see (16.2.1) and §15.2(i).

28: 15.4 Special Cases

… ►

§15.4(i) Elementary Functions

… ► ►

§15.4(ii) Argument Unity

… ►

Chu–Vandermonde Identity

… ►

§15.4(iii) Other Arguments

…

29: 13.27 Mathematical Applications

§13.27 Mathematical Applications

►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … ►

13.27.1

ⓘ

Permalink:

http://dlmf.nist.gov/13.27.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.27 and Ch.13

►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. … …

30: 15.9 Relations to Other Functions

… ►

§15.9(i) Orthogonal Polynomials

… ►

Jacobi

… ►

Legendre

… ►

Meixner

… ►

Meixner–Pollaczek

…