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21: 15.10 Hypergeometric Differential Equation

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15.10.8 $$F(\genfrac{}{}{0pt}{}{a,b}{n};z)\ln z-\sum _{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!{\left(1-a\right)}_k{\left(1-b\right)}_k}{(-z)}^{-k}+\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{{\left(n\right)}_{k}k!}{z}^k\left(\psi \left(a+k\right)+\psi \left(b+k\right)-\psi \left(1+k\right)-\psi \left(n+k\right)\right),$$ $a,b\ne n-1,n-2,\mathrm{\dots },0,-1,-2,\mathrm{\dots }$,

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Symbols:

$F(a,b;c;z)$ 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), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $k$: integer and $m$: integer

A&S Ref:

15.5.19. (Compared with (15.10.8) this result has a multiple of $F(a,b;n;z)$ added to the right-hand side. It also contains an error: the conditions on $a$ and $b$ should be $a,b\ne 1,2,\mathrm{\dots },m$.)

Referenced by:

(15.10.8)

Permalink:

http://dlmf.nist.gov/15.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.9 $$F(\genfrac{}{}{0pt}{}{-m,b}{n};z)\ln z-\sum _{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!{\left(m+1\right)}_k{\left(1-b\right)}_k}{(-z)}^{-k}+\sum _{k=0}^m\frac{{\left(-m\right)}_k{\left(b\right)}_k}{{\left(n\right)}_{k}k!}{z}^k\left(\psi \left(1+m-k\right)+\psi \left(b+k\right)-\psi \left(1+k\right)-\psi \left(n+k\right)\right)+{(-1)}^{m}m!\sum _{k=m+1}^{\mathrm{\infty }}\frac{(k-1-m)!{\left(b\right)}_k}{{\left(n\right)}_{k}k!}{z}^k,$$ $a=-m$, $m=0,1,2,\mathrm{\dots }$; $b\ne n-1,n-2,\mathrm{\dots },0,-1,-2,\mathrm{\dots }$,

ⓘ

Symbols:

$F(a,b;c;z)$ 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), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $k$: integer and $m$: integer

Referenced by:

§15.10(i), §15.10(i)

Permalink:

http://dlmf.nist.gov/15.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.10 $$F(\genfrac{}{}{0pt}{}{-m,-\mathrm{\ell }}{n};z)\ln z-\sum _{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!{\left(m+1\right)}_k{\left(\mathrm{\ell }+1\right)}_k}{(-z)}^{-k}+\sum _{k=0}^{\mathrm{\ell }}\frac{{\left(-m\right)}_k{\left(-\mathrm{\ell }\right)}_k}{{\left(n\right)}_{k}k!}{z}^k\left(\psi \left(1+m-k\right)+\psi \left(1+\mathrm{\ell }-k\right)-\psi \left(1+k\right)-\psi \left(n+k\right)\right)+{(-1)}^{\mathrm{\ell }}\mathrm{\ell }!\sum _{k=\mathrm{\ell }+1}^m\frac{(k-1-\mathrm{\ell })!{\left(-m\right)}_k}{{\left(n\right)}_{k}k!}{z}^k,$$ $a=-m$, $m=0,1,2,\mathrm{\dots }$; $b=-\mathrm{\ell }$, $\mathrm{\ell }=0,1,2,\mathrm{\dots },m$.

ⓘ

Symbols:

$F(a,b;c;z)$ 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), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $k$: integer, $\mathrm{\ell }$: integer and $m$: integer

Referenced by:

§15.10(i)

Permalink:

http://dlmf.nist.gov/15.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

… ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

22: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

23: 23 Weierstrass Elliptic and Modular
Functions

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24: 16.1 Special Notation

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$p,q$	nonnegative integers.
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${\left(𝐚\right)}_k$	${\left(a_1\right)}_k{\left(a_2\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k$.
${\left(𝐛\right)}_k$	${\left(b_1\right)}_k{\left(b_2\right)}_k\mathrm{\cdots }{\left(b_q\right)}_k$.
…

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25: 16.9 Zeros

… ►Next, assume that $p=q$ and that the $a_j$ and the quotients ${\left(𝐚\right)}_j/{\left(𝐛\right)}_j$ are all real. …

26: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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17.9.1 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=\frac{{(za;q)}_{\mathrm{\infty }}}{{(z;q)}_{\mathrm{\infty }}}{}_2{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,c/b}{c,az};q,bz),$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

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17.9.3 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=\frac{{(abz/c;q)}_{\mathrm{\infty }}}{{(bz/c;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,c/b,0}{c,cq/(bz)};q,q)+\frac{{(a,bz,c/b;q)}_{\mathrm{\infty }}}{{(c,z,c/(bz);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{z,abz/c,0}{bz,bzq/c};q,q),$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Referenced by:

Erratum (V1.0.23) for Equation (17.9.3)

Permalink:

http://dlmf.nist.gov/17.9.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${}_2{}^{}\varphi _1^{}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).

Suggested 2019-04-26 by Roberto S. Costas-Santos

See also:

Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

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17.9.4 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},b}{c};q,z)=\frac{{(c/b;q)}_n}{{(c;q)}_n}{\left(\frac{bz}{q}\right)}^n{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},q/z,q^{1-n}/c}{b{q}^{1-n}/c,0};q,q),$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

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17.9.19 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(a;q^2)}_n{(b;q)}_n}{{(q^2;q^2)}_n{(c;q)}_n}{z}^n=\frac{{(b;q)}_{\mathrm{\infty }}{(az;q^2)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(z;q^2)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_{2n}{(z;q^2)}_n{b}^{2n}}{{(q;q)}_{2n}{(az;q^2)}_n}+\frac{{(b;q)}_{\mathrm{\infty }}{(azq;q^2)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(zq;q^2)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_{2n+1}{(zq;q^2)}_n{b}^{2n+1}}{{(q;q)}_{2n+1}{(azq;q^2)}_n}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

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17.9.20 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(a;q^k)}_n{(b;q)}_{kn}{z}^n}{{(q^k;q^k)}_n{(c;q)}_{kn}}=\frac{{(b;q)}_{\mathrm{\infty }}{(az;q^k)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(z;q^k)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_n{(z;q^k)}_n{b}^n}{{(q;q)}_n{(az;q^k)}_n},$$ $k=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

27: 17.12 Bailey Pairs

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17.12.3 $${\beta }_n=\sum _{j=0}^n\frac{{\alpha }_j}{{(q;q)}_{n-j}{(aq;q)}_{n+j}}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

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17.12.4 $$\sum _{n=0}^{\mathrm{\infty }}{q}^{n^2}{a}^n{\beta }_n=\frac{1}{{(aq;q)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}{q}^{n^2}{a}^n{\alpha }_n.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

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$${(\frac{aq}{{\rho }_1},\frac{aq}{{\rho }_2};q)}_n{\beta }_n^{\prime }=\sum _{j=0}^n{({\rho }_1,{\rho }_2;q)}_j{(\frac{aq}{{\rho }_1{\rho }_2};q)}_{n-j}{\left(\frac{aq}{{\rho }_1{\rho }_2}\right)}^j\frac{{\beta }_j}{{(q;q)}_{n-j}}$$

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${\alpha }_n={\displaystyle \frac{{(a;q)}_n(1-a{q}^{2n}){(-1)}^n{q}^{n(3n-1)/2}{a}^n}{{(q;q)}_n(1-a)}},$

►

${\beta }_n={\displaystyle \frac{1}{{(q;q)}_n}}.$

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28: 18.25 Wilson Class: Definitions

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18.25.11 $${\omega }_y=\frac{{\left(\alpha +1\right)}_y{\left(\beta +\delta +1\right)}_y{\left(\gamma +1\right)}_y{\left(\gamma +\delta +2\right)}_y}{{\left(-\alpha +\gamma +\delta +1\right)}_y{\left(-\beta +\gamma +1\right)}_y{\left(\delta +1\right)}_{y}y!},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $y$: real variable and $\delta$: arbitrary small positive constant

Referenced by:

§18.25(i)

Permalink:

http://dlmf.nist.gov/18.25.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

►

18.25.12 $$h_n=\frac{{\left(-\beta \right)}_N{\left(\gamma +\delta +2\right)}_N}{{\left(-\beta +\gamma +1\right)}_N{\left(\delta +1\right)}_N}\frac{{\left(n+\alpha +\beta +1\right)}_{n}n!}{{\left(\alpha +\beta +2\right)}_{2n}}\frac{{\left(\alpha +\beta -\gamma +1\right)}_n{\left(\alpha -\delta +1\right)}_n{\left(\beta +1\right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +\delta +1\right)}_n{\left(\gamma +1\right)}_n}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $h_n$

Referenced by:

§18.25(iii), §18.38(iii)

Permalink:

http://dlmf.nist.gov/18.25.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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18.25.14 $${\omega }_y=\frac{{(-1)}^y{\left(-N\right)}_y{\left(\gamma +1\right)}_y{\left(\gamma +\delta +1\right)}_2}{{\left(N+\gamma +\delta +2\right)}_y{\left(\delta +1\right)}_{y}y!},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $y$: real variable, $N$: positive integer and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/18.25.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

►

18.25.15 $$h_n=\frac{n!(N-n)!{\left(\gamma +\delta +2\right)}_N}{N!{\left(\gamma +1\right)}_n{\left(\delta +1\right)}_{N-n}}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $h_n$

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

… ►

Table 18.25.2: Wilson class OP’s: leading coefficients.

► ►►►

$p_n(x)$	$k_n$
…
$R_n(x;\alpha ,\beta ,\gamma ,\delta )$	${\displaystyle \frac{{\left(n+\alpha +\beta +1\right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +\delta +1\right)}_n{\left(\gamma +1\right)}_n}}$
…

►

29: 13.7 Asymptotic Expansions for Large Argument

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13.7.1 $$𝐌(a,b,x)\sim \frac{{\mathrm{e}}^x{x}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{x}^{-s},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(i), §13.7 and Ch.13

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13.7.3 $$U(a,b,z)\sim z^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

13.5.2

Referenced by:

§12.9(i), §13.12, §13.2(i)

Permalink:

http://dlmf.nist.gov/13.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(i), §13.7 and Ch.13

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13.7.4 $$U(a,b,z)=z^{-a}\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s}+{\epsilon }_n(z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and ${\epsilon }_n(z)$: function

Referenced by:

§13.2(i), §13.7(ii)

Permalink:

http://dlmf.nist.gov/13.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(ii), §13.7 and Ch.13

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13.7.10 $$U(a,b,z)=z^{-a}\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s}+R_n(a,b,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and $R_n(a,b,z)$: remainder

Referenced by:

§13.29(i)

Permalink:

http://dlmf.nist.gov/13.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

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13.7.11 $$R_n(a,b,z)=\frac{{(-1)}^{n}2\pi z^{a-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}\left(\sum _{s=0}^{m-1}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{(-z)}^{-s}{G}_{n+2a-b-s}(z)+{\left(1-a\right)}_m{\left(b-a\right)}_m{R}_{m,n}(a,b,z)\right),$$

ⓘ

Defines:

$R_n(a,b,z)$: remainder (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and $G_p(z)$: expansion

Referenced by:

§13.29(i)

Permalink:

http://dlmf.nist.gov/13.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

…

30: 13.19 Asymptotic Expansions for Large Argument

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13.19.1 $$M_{\kappa ,\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}x}{x}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{x}^{-s},$$ $\mu -\kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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13.19.2 $$M_{\kappa ,\mu }\left(z\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{z}^{-s}+\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{-\frac{1}{2}z\pm (\frac{1}{2}+\mu -\kappa )\pi \mathrm{i}}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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