Pochhammer symbol

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41: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

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16.10.1 $${}_{p+r}{}^{}F_{q+s}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,c_1,\mathrm{\dots },c_r}{b_1,\mathrm{\dots },b_q,d_1,\mathrm{\dots },d_s};z\zeta )=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(𝐚\right)}_k{\left(\alpha \right)}_k{\left(\beta \right)}_k{(-z)}^k}{{\left(𝐛\right)}_k{\left(\gamma +k\right)}_{k}k!}{}_{p+2}{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{\alpha +k,\beta +k,a_1+k,\mathrm{\dots },a_p+k}{\gamma +2k+1,b_1+k,\mathrm{\dots },b_q+k};z){}_{r+2}{}^{}F_{s+2}^{}(\genfrac{}{}{0pt}{}{-k,\gamma +k,c_1,\mathrm{\dots },c_r}{\alpha ,\beta ,d_1,\mathrm{\dots },d_s};\zeta ).$$

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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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: 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

Permalink:

http://dlmf.nist.gov/16.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

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16.10.2 $${}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};z\zeta )={(1-z)}^{-a_1}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k}{k!}{}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{-k,a_2,\mathrm{\dots },a_{p+1}}{b_1,\mathrm{\dots },b_p};\zeta ){\left(\frac{z}{z-1}\right)}^k.$$

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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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $z$: 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

Permalink:

http://dlmf.nist.gov/16.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.10 and Ch.16

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42: 26.8 Set Partitions: Stirling Numbers

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26.8.7 $$\sum _{k=0}^{n}s(n,k)x^k={\left(x-n+1\right)}_n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $s(n,k)$: Stirling number of the first kind, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.3 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.8.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

The Pochhammersymbol now links to its definition.

See also:

Annotations for §26.8(ii), §26.8 and Ch.26

►where ${\left(x\right)}_n$ is the Pochhammer symbol: $x(x+1)\mathrm{\cdots }(x+n-1)$. … ►

26.8.10 $$\sum _{k=1}^{n}S(n,k){\left(x-k+1\right)}_k=x^n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $S(n,k)$: Stirling number of the second kind, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.4

Referenced by:

§26.15

Permalink:

http://dlmf.nist.gov/26.8.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

The Pochhammersymbol now links to its definition.

See also:

Annotations for §26.8(ii), §26.8 and Ch.26

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43: 31.11 Expansions in Series of Hypergeometric Functions

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31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

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Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3_2 $$P_j^6=\frac{{\left(\lambda -\mu \right)}_{2j}}{{\left(1-\mu \right)}_j{\left(\gamma -\mu \right)}_j}{z}^{-\mu +j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\mu -j,1-\gamma +\mu -j}{1-\lambda +\mu -2j};\frac{1}{z}).$$

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Defines:

$P_j^6$: second Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.12 $$P_j^5=\frac{{\left(\alpha \right)}_j{\left(1-\gamma +\alpha \right)}_j}{{\left(1+\alpha -\beta +ϵ\right)}_{2j}}{z}^{-\alpha -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +j,1-\gamma +\alpha +j}{1+\alpha -\beta +ϵ+2j};\frac{1}{z}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.1.7):

We have rewritten the gamma functions in the prefactor more concisely using Pochhammersymbols.

See also:

Annotations for §31.11(iii), §31.11 and Ch.31

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44: 15.8 Transformations of Variable

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15.8.6 $$F(\genfrac{}{}{0pt}{}{-m,b}{c};z)=\frac{{\left(b\right)}_m}{{\left(c\right)}_m}{(-z)}^{m}F(\genfrac{}{}{0pt}{}{-m,1-c-m}{1-b-m};\frac{1}{z})=\frac{{\left(b\right)}_m}{{\left(c\right)}_m}{(1-z)}^{m}F(\genfrac{}{}{0pt}{}{-m,c-b}{1-b-m};\frac{1}{1-z}),$$

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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), $z$: complex variable, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

Referenced by:

§15.8(ii), §15.8(ii), §18.17(vii)

Permalink:

http://dlmf.nist.gov/15.8.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammersymbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

►

15.8.7 $$F(\genfrac{}{}{0pt}{}{-m,b}{c};z)=\frac{{\left(c-b\right)}_m}{{\left(c\right)}_m}F(\genfrac{}{}{0pt}{}{-m,b}{b-c-m+1};1-z)=\frac{{\left(c-b\right)}_m}{{\left(c\right)}_m}{z}^{m}F(\genfrac{}{}{0pt}{}{-m,1-c-m}{b-c-m+1};1-\frac{1}{z}),$$

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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), $z$: complex variable, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

Referenced by:

§15.19(iv), §15.8(ii), §15.8(ii), (18.35.4_5)

Permalink:

http://dlmf.nist.gov/15.8.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammersymbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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15.8.9 $$𝐅(\genfrac{}{}{0pt}{}{a,a+m}{c};z)=\frac{{(1-z)}^{-a}}{\mathrm{\Gamma }\left(a+m\right)\mathrm{\Gamma }\left(c-a\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k{\left(c-a-m\right)}_k(m-k-1)!}{k!}{(z-1)}^{-k}+\frac{{(-1)}^m{(1-z)}^{-a-m}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-a-m\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k{\left(c-a\right)}_k}{k!(k+m)!}{(1-z)}^{-k}(\ln \left(1-z\right)+\psi \left(k+1\right)+\psi \left(k+m+1\right)-\psi \left(a+k+m\right)-\psi \left(c-a+k\right)),$$ .

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

Referenced by:

§15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammersymbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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15.8.10 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b+m};z)=\frac{1}{\mathrm{\Gamma }\left(a+m\right)\mathrm{\Gamma }\left(b+m\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k{\left(b\right)}_k(m-k-1)!}{k!}{(z-1)}^k-\frac{{(z-1)}^m}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k{\left(b+m\right)}_k}{k!(k+m)!}{(1-z)}^k\left(\ln \left(1-z\right)-\psi \left(k+1\right)-\psi \left(k+m+1\right)+\psi \left(a+k+m\right)+\psi \left(b+k+m\right)\right),$$ ,

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

Referenced by:

§15.4(ii), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammersymbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

►

15.8.11 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b+m};z)=\frac{z^{-a}}{\mathrm{\Gamma }\left(a+m\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k(m-k-1)!}{k!\mathrm{\Gamma }\left(b+m-k\right)}{\left(1-\frac{1}{z}\right)}^k-\frac{z^{-a}}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k}{k!(k+m)!\mathrm{\Gamma }\left(b-k\right)}{(-1)}^k{\left(1-\frac{1}{z}\right)}^{k+m}\left(\ln \left(\frac{1-z}{z}\right)-\psi \left(k+1\right)-\psi \left(k+m+1\right)+\psi \left(a+k+m\right)+\psi \left(b-k\right)\right),$$ .

ⓘ

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

Referenced by:

§15.8(ii), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammersymbols now link to their definition.

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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45: 13.2 Definitions and Basic Properties

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13.2.3 $$𝐌(a,b,z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{\mathrm{\Gamma }\left(b+s\right)s!}{z}^s,$$

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Defines:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i)

Permalink:

http://dlmf.nist.gov/13.2.E3

Encodings:

pMML, png, TeX

See also:

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

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13.2.5 $$\underset{b\to -n}{lim}\frac{M(a,b,z)}{\mathrm{\Gamma }\left(b\right)}=𝐌(a,-n,z)=\frac{{\left(a\right)}_{n+1}}{(n+1)!}{z}^{n+1}M(a+n+1,n+2,z).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E5

Encodings:

pMML, png, TeX

See also:

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

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13.2.14 $$U(-n,b,z)={(-1)}^n{\left(b\right)}_n+O\left(z\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E14

Encodings:

pMML, png, TeX

See also:

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

►

13.2.15 $$U(-n+b-1,b,z)={(-1)}^n{\left(2-b\right)}_n{z}^{1-b}+O\left(z^{2-b}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E15

Encodings:

pMML, png, TeX

See also:

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

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13.2.27 $$\sum _{k=1}^n\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)}_k}{z}^{-k}-\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(n+1\right)}_{k}k!}{z}^k\left(\ln z+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(n+k+1\right)\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: 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, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(v)

Permalink:

http://dlmf.nist.gov/13.2.E27

Encodings:

pMML, png, TeX

See also:

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

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46: 13.15 Recurrence Relations and Derivatives

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13.15.15 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(z\right)\right)={(-1)}^n{\left(-2\mu \right)}_n{\mathrm{e}}^{\frac{1}{2}z}{z}^{\mu -\frac{1}{2}(n+1)}{M}_{\kappa -\frac{1}{2}n,\mu -\frac{1}{2}n}\left(z\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

►

13.15.16 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(z\right)\right)=\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_n}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\mu -\frac{1}{2}(n+1)}{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(z\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

►

13.15.17 $${\left(z\frac{d}{dz}z\right)}^n\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa -1}{M}_{\kappa ,\mu }\left(z\right)\right)={\left(\frac{1}{2}+\mu -\kappa \right)}_n{\mathrm{e}}^{\frac{1}{2}z}{z}^{n-\kappa -1}{M}_{\kappa -n,\mu }\left(z\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

… ►

13.15.20 $${\left(z\frac{d}{dz}z\right)}^n\left({\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa -1}{M}_{\kappa ,\mu }\left(z\right)\right)={\left(\frac{1}{2}+\mu +\kappa \right)}_n{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa +n-1}{M}_{\kappa +n,\mu }\left(z\right).$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

… ►

13.15.23 $${\left(z\frac{d}{dz}z\right)}^n\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa -1}{W}_{\kappa ,\mu }\left(z\right)\right)={\left(\frac{1}{2}+\mu -\kappa \right)}_n{\left(\frac{1}{2}-\mu -\kappa \right)}_n{\mathrm{e}}^{\frac{1}{2}z}{z}^{n-\kappa -1}{W}_{\kappa -n,\mu }\left(z\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(ii), §13.15 and Ch.13

…

47: 3.9 Acceleration of Convergence

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3.9.16 $$c_{j,k,n}=\frac{{\left(\beta +n+j\right)}_{k-1}}{{\left(\beta +n+k\right)}_{k-1}},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $c_{j,k,n}$: coefficient and $\beta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/3.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(v), §3.9 and Ch.3

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3.9.17 $$c_{j,k,n}=\frac{{\left(-\gamma -n-j\right)}_{k-1}}{{\left(-\gamma -n-k\right)}_{k-1}},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\gamma$: arbitrary constant and $c_{j,k,n}$: coefficient

Permalink:

http://dlmf.nist.gov/3.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(v), §3.9 and Ch.3

►where ${\left(a\right)}_0=1$ and ${\left(a\right)}_j=a(a+1)\mathrm{\cdots }(a+j-1)$ are Pochhammer symbols (§5.2(iii)), and the constants $\beta$ and $\gamma$ are chosen arbitrarily subject to certain conditions. …

48: 13.29 Methods of Computation

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13.29.3 $${\mathrm{e}}^{-\frac{1}{2}z}=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(2\mu \right)}_s{\left(\frac{1}{2}+\mu -\kappa \right)}_s}{{\left(2\mu \right)}_{2s}s!}{(-z)}^{s}y(s),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable and $y(n)$: solution

Permalink:

http://dlmf.nist.gov/13.29.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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13.29.6 $$w(n)={\left(a\right)}_{n}U(n+a,b,z),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.29.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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13.29.7 $$z^{-a}=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a-b+1\right)}_s}{s!}w(s),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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49: 18.12 Generating Functions

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18.12.3 $${(1+z)}^{-\alpha -\beta -1}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(\alpha +\beta +1),\frac{1}{2}(\alpha +\beta +2)}{\beta +1};\frac{2(x+1)z}{{(1+z)}^2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\alpha +\beta +1\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ ,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.12.3_5), §18.12, §18.12

Permalink:

http://dlmf.nist.gov/18.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.3_5 $$\frac{1+z}{{(1-2xz+z^2)}^{\beta +\frac{3}{2}}}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\beta +2\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\beta +1,\beta )}\left(x\right)z^n,$$ ,

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Substitute $\alpha =\beta +1$ in (18.12.3) and use (15.4.6).

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.4 $${(1-2xz+z^2)}^{-\lambda }=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\lambda )}\left(x\right)z^n=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right)z^n,$$ .

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.3

Referenced by:

§18.12, §18.18(viii)

Permalink:

http://dlmf.nist.gov/18.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.6 $$\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right){\mathrm{e}}^{z\cos \theta }{(\frac{1}{2}z\sin \theta )}^{\frac{1}{2}-\lambda }{J}_{\lambda -\frac{1}{2}}\left(z\sin \theta \right)=\sum _{n=0}^{\mathrm{\infty }}\frac{C_n^{(\lambda )}\left(\cos \theta \right)}{{\left(2\lambda \right)}_n}{z}^n,$$ $0\le \theta \le \pi$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.14 $$\mathrm{\Gamma }\left(\alpha +1\right){(xz)}^{-\frac{1}{2}\alpha }{\mathrm{e}}^z{J}_{\alpha }\left(2\sqrt{xz}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(\alpha )}\left(x\right)}{{\left(\alpha +1\right)}_n}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.16

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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50: 8.7 Series Expansions

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8.7.2 $$\gamma (a,x+y)-\gamma (a,x)=\mathrm{\Gamma }(a,x)-\mathrm{\Gamma }(a,x+y)={\mathrm{e}}^{-x}{x}^{a-1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{{(-x)}^n}(1-{\mathrm{e}}^{-y}{e}_n(y)),$$ .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.30

Permalink:

http://dlmf.nist.gov/8.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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8.7.5 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{\mathrm{\Gamma }\left(n+a+1\right)}\left(2n+1\right){𝗂}_n^{(1)}\left(\frac{1}{2}z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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