Pochhammer symbol

SearchAdvancedHelp

(0.008 seconds)

21—30 of 125 matching pages

21: 17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

… ►

17.10.1 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,d/a,c/b,dq/(abz);q)}_{\mathrm{\infty }}}{{(z,d,q/b,cd/(abz);q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,abz/d}{az,c};q,\frac{d}{a}),$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.2 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,bz,cq/(abz),dq/(abz);q)}_{\mathrm{\infty }}}{{(q/a,q/b,c,d;q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{abz/c,abz/d}{az,bz};q,\frac{cd}{abz}).$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.3 $${}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},c,d,e,f,a{q}^{-n},q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},a{q}^{n+1}};q,\frac{a^2{q}^{2n+2}}{cdef})=\frac{{(aq,q/a,aq/(cd),aq/(ef);q)}_n}{{(q/c,q/d,aq/e,aq/f;q)}_n}{}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{e,f,a{q}^{n+1}/(cd),q^{-n}}{aq/c,aq/d,q^{n+1},ef/(a{q}^n)};q,q),$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.4 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{e,f}{aq/c,aq/d};q,\frac{aq}{ef})=\frac{{(q/c,q/d,aq/e,aq/f;q)}_{\mathrm{\infty }}}{{(aq,q/a,aq/(cd),aq/(ef);q)}_{\mathrm{\infty }}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(1-a{q}^{2n}){(c,d,e,f;q)}_n}{(1-a){(aq/c,aq/d,aq/e,aq/f;q)}_n}{\left(\frac{q{a}^3}{cdef}\right)}^n{q}^{n^2}.$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.5 $$\frac{{(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q)}_{\mathrm{\infty }}}{{(fa,ga,f/a,g/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^2}{bcdefg})=\frac{{(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf/e;q)}_{\mathrm{\infty }}}{{(fa,q/(fa),aq/f,f/a,g/f,fg,q{f}^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{f^2,qf,-qf,fb,fc,fd,fe,fg}{f,-f,fq/b,fq/c,fq/d,fq/e,fq/g};q,\frac{q^2}{bcdefg})+\mathrm{idem}(f;g).$$

ⓘ

Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

…

22: 26.17 The Twelvefold Way

… ►In this table ${\left(k\right)}_n$ is Pochhammer’s symbol, and $S(n,k)$ and $p_k\left(n\right)$ are defined in §§26.8(i) and 26.9(i). …

23: 26.15 Permutations: Matrix Notation

… ►

26.15.11 $$\sum _{k=0}^n{r}_{n-k}(B){\left(x-k+1\right)}_k=\prod _{j=1}^n(x+b_j-j+1).$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

The Pochhammersymbol now links to its definition.

See also:

Annotations for §26.15, §26.15 and Ch.26

… ►

26.15.12 $$\sum _{k=0}^n{r}_{n-k}(B){\left(x-k+1\right)}_k=x^n,$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E12

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

The Pochhammersymbol now links to its definition.

See also:

Annotations for §26.15, §26.15 and Ch.26

…

24: 10.8 Power Series

… ►

10.8.3 $$J_{\nu }\left(z\right)J_{\mu }\left(z\right)={(\frac{1}{2}z)}^{\nu +\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\nu +\mu +k+1\right)}_k{(-\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\mu +k+1\right)}.$$

ⓘ

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), $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

Proof sketch:

Rewrite (16.12.1).

A&S Ref:

9.1.14

Referenced by:

§10.22(ii), §10.31, §10.8, §10.8, §18.17(v), Erratum (V1.0.17) for Section 10.8

Permalink:

http://dlmf.nist.gov/10.8.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammersymbol now links to its definition.

See also:

Annotations for §10.8 and Ch.10

…

25: 7.12 Asymptotic Expansions

… ►

7.12.2 $$\mathrm{f}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}},$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.27

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E2

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

►

7.12.3 $$\mathrm{g}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi z^2/2)}^{2m+1}},$$

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.28

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E3

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{{\pi }^2{z}^3}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m+1)}{{(\pi z^2)}^{2m}}$ We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

… ►

7.12.4 $$\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}}+R_n^{(\mathrm{f})}(z),$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

A&S Ref:

7.3.27 (in different form)

Referenced by:

§7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E4

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{f})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

►

7.12.5 $$\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi z^2/2)}^{2m+1}},+R_n^{(\mathrm{g})}(z),$$

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{g})}(z)$: remainder term

A&S Ref:

7.3.28 (in different form)

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E5

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{{\pi }^2{z}^3}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m+1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{g})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

…

26: 13.13 Addition and Multiplication Theorems

… ►

13.13.1 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{y}^n}{{\left(b\right)}_{n}n!}M(a+n,b+n,x),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/13.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

►

13.13.2 $${\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-b\right)}_n{(-y/x)}^n}{n!}M(a,b-n,x),$$ ,

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/13.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

… ►

13.13.4 $${\mathrm{e}}^y\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(b-a\right)}_n{(-y)}^n}{{\left(b\right)}_{n}n!}M(a,b+n,x),$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(i), §13.13 and Ch.13

… ►

13.13.7 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_n{(-y)}^n}{n!}U(a+n,b+n,x),$$ ,

ⓘ

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, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

►

13.13.8 $${\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1+a-b\right)}_n{(-y/x)}^n}{n!}U(a,b-n,x),$$ ,

ⓘ

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, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

…

27: 16.13 Appell Functions

… ►

16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_m{\left({\alpha }^{\prime }\right)}_n{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

ⓘ

Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

…

28: 13.26 Addition and Multiplication Theorems

… ►

13.26.2 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{y}{\sqrt{x}}\right)}^n{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/13.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

►

13.26.3 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa -n,\mu }\left(x\right),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

ⓘ

Symbols:

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

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

… ►

13.26.5 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{-y}{\sqrt{x}}\right)}^n{M}_{\kappa +\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/13.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

►

13.26.6 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x}{x+y}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa +n,\mu }\left(x\right),$$ $\mathrm{\Re }\left((y+x)/x\right)>\frac{1}{2}$.

ⓘ

Symbols:

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

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

… ►

13.26.9 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n{\left(\frac{1}{2}-\mu -\kappa \right)}_n}{n!}{\left(\frac{y}{x+y}\right)}^n{W}_{\kappa -n,\mu }\left(x\right),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(ii), §13.26 and Ch.13

…

29: 12.9 Asymptotic Expansions for Large Variable

… ►

12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

ⓘ

Symbols:

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

A&S Ref:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

ⓘ

Symbols:

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

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

…

30: 25.8 Sums

… ►

25.8.3 $$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_k\zeta \left(s+k\right)}{k!2^{s+k}}=(1-2^{-s})\zeta \left(s\right),$$ $s\ne 1$.

ⓘ

Symbols:

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

Keywords:

infinite series

Source:

Srivastava (1988, (2.6), p. 131); with $k=0$ term

Referenced by:

§25.11(iv), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.8.E3

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)\zeta \left(s+k\right)}{k!\mathrm{\Gamma }\left(s\right)2^{s+k}}$ more concisely as ${\sum }_{k=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_k\zeta \left(s+k\right)}{k!2^{s+k}}$ using the Pochhammersymbol.

See also:

Annotations for §25.8 and Ch.25

…