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11: 36.2 Catastrophes and Canonical Integrals

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36.2.1 $${\mathrm{\Phi }}_K(t;𝐱)=t^{K+2}+\sum _{m=1}^K{x}_m{t}^m.$$

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Defines:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$

Symbols:

$n$: integer, $t$: variable, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Referenced by:

§36.10(i), §36.12(i), §36.5(ii), §36.7(ii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.2 $${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)=s^3-3s{t}^2+z(s^2+t^2)+yt+xs,$$ $𝐱=\{x,y,z\}$,

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Defines:

${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$: elliptic umbilic catastrophe

Symbols:

$y$: real parameter, $z$: real parameter, $t$: variable, $s$: variable and $x$: real parameter

Referenced by:

§36.10(iii), §36.2(i), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.3 $${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)=s^3+t^3+zst+yt+xs,$$ $𝐱=\{x,y,z\}$,

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Defines:

${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$: hyperbolic umbilic catastrophe

Symbols:

$y$: real parameter, $z$: real parameter, $t$: variable, $s$: variable and $x$: real parameter

Referenced by:

§36.10(iii), §36.2(i), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.10 $${\mathrm{\Psi }}_K(𝐱;k)=\sqrt{k}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}k{\mathrm{\Phi }}_K(t;𝐱)\right)dt,$$ $k>0$.

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Defines:

${\mathrm{\Psi }}_K(𝐱;k)$: diffraction catastrophe

Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: variable, $t$: variable and $K$: codimension

Referenced by:

§36.12(i)

Permalink:

http://dlmf.nist.gov/36.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

… ►For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975). …

12: 5.2 Definitions

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5.2.5 $${\left(a\right)}_n=\mathrm{\Gamma }\left(a+n\right)/\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

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

A&S Ref:

6.1.22

Referenced by:

(25.11.28), (25.5.7), (25.8.2)

Permalink:

http://dlmf.nist.gov/5.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

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5.2.6 $${\left(-a\right)}_n={(-1)}^n{\left(a-n+1\right)}_n,$$

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Symbols:

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

Referenced by:

§5.2(iii), Erratum (V1.0.17) for Subsection 5.2(iii)

Permalink:

http://dlmf.nist.gov/5.2.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation was added.

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

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13: 1.9 Calculus of a Complex Variable

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1.9.63 $$f^{(m)}(z)=\sum _{n=0}^{\mathrm{\infty }}{\left(n+1\right)}_m{a}_{n+m}{(z-z_0)}^n,$$ , $m=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: variable, $m$: nonnegative integer, $n$: nonnegative integer, $R$: radius of convergence and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.9.E63

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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14: 18.28 Askey–Wilson Class

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18.28.4 $$h_0=\frac{{(abcd;q)}_{\mathrm{\infty }}}{{(q,ab,ac,ad,bc,bd,cd;q)}_{\mathrm{\infty }}},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable and $h_n$

Permalink:

http://dlmf.nist.gov/18.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.17 $$\frac{1}{2\pi }{\int }_0^{\pi }{H}_n\left(\cos \theta |q\right)H_m\left(\cos \theta |q\right){\left|{({\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1};q)}_{\mathrm{\infty }}}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.28.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(vi), §18.28 and Ch.18

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18.28.25 $$P_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}(x|q)=\frac{q^{\frac{1}{2}n\lambda }{(q^{\lambda +1/2};q)}_n}{{(q^{2\lambda };q)}_n}{C}_n(x;q^{\lambda }|q).$$

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Symbols:

$(a,b)$: open interval, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper and Rahman (2004, (7.5.33))(proved)

Permalink:

http://dlmf.nist.gov/18.28.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(ix), §18.28 and Ch.18

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18.28.27 $$\underset{\lambda \to 0}{lim}{r}_n(bqx/(2\lambda );\lambda ,qb{\lambda }^{-1},q,a|q)={(-b)}^n{q}^{n(n+1)/2}\frac{{(qa;q)}_n}{{(qb;q)}_n}{p}_n(x;a,b;q).$$

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Symbols:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1993, (6.4)); The limit formula there is equivalent to the present formula(proved)

Proof sketch:

Proceed in a similar way as in the proof of (18.28.26). After taking the limit one now obtains a ${}_3{}^{}\varphi _2^{}$ which equals (18.27.14_1) by (18.27.13).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

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18.28.32 $$\underset{\beta \to 0}{lim}{C}_n(x;\beta |q)=\frac{H_n\left(x|q\right)}{{(q;q)}_n}.$$

ⓘ

Symbols:

$H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.10.34))

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

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15: 15.8 Transformations of Variable

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

ⓘ

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):

Pochhammer symbols now link to their definition.

See also:

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

►

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

ⓘ

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):

Pochhammer symbols now link to their definition.

See also:

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

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15.8.8 $$𝐅(\genfrac{}{}{0pt}{}{a,a+m}{c};z)=\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(a+m\right)}\sum _{k=0}^{m-1}\frac{{\left(a\right)}_k(m-k-1)!}{k!\mathrm{\Gamma }\left(c-a-k\right)}{z}^{-k}+\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+m\right)}_k}{k!(k+m)!\mathrm{\Gamma }\left(c-a-k-m\right)}{(-1)}^k{z}^{-k-m}\left(\ln \left(-z\right)+\psi \left(k+1\right)+\psi \left(k+m+1\right)-\psi \left(a+k+m\right)-\psi \left(c-a-k-m\right)\right),$$ ,

ⓘ

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, $𝐅(a,b;c;z)$ 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:

§15.12(i), §15.19(iv), §15.8(ii), §15.8(ii), §16.11(i)

Permalink:

http://dlmf.nist.gov/15.8.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

Pochhammer symbols now link to their definition.

See also:

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

►

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

ⓘ

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):

Pochhammer symbols now link to their definition.

See also:

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

… ►

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

ⓘ

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):

Pochhammer symbols now link to their definition.

See also:

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

…

16: 17.14 Constant Term Identities

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17.14.1

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Symbols:

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

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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17.14.2 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}q;q)}_{\mathrm{\infty }}}=\frac{H(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $H(q)$: LHS of (17.2.50)

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

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17.14.3 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q)}_{\mathrm{\infty }}}=\frac{G(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

►

17.14.4 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2}}{{(q^2;q^2)}_n{(q;q^2)}_n}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-z^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1};q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-2};q^4)}_{\mathrm{\infty }}}=\frac{G(q^4)}{{(q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

►

17.14.5 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2+2n}}{{(q^2;q^2)}_n{(q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-q^2{z}^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(q^4{z}^{-2};q^4)}_{\mathrm{\infty }}}=\frac{H(q^4)}{{(q;q^2)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $H(q)$: LHS of (17.2.50)

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

…

17: 15.16 Products

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15.16.1 $$F(\genfrac{}{}{0pt}{}{a,b}{c-\frac{1}{2}};z)F(\genfrac{}{}{0pt}{}{c-a,c-b}{c+\frac{1}{2}};z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(c\right)}_s}{{\left(c+\frac{1}{2}\right)}_s}{A}_s{z}^s,$$ ,

ⓘ

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), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $A_s$: coefficients

Permalink:

http://dlmf.nist.gov/15.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

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15.16.3 $$F(\genfrac{}{}{0pt}{}{a,b}{c};z)F(\genfrac{}{}{0pt}{}{a,b}{c};\zeta )=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s{\left(c-a\right)}_s{\left(c-b\right)}_s}{{\left(c\right)}_s{\left(c\right)}_{2s}s!}{\left(z\zeta \right)}^{s}F(\genfrac{}{}{0pt}{}{a+s,b+s}{c+2s};z+\zeta -z\zeta ),$$ , , .

ⓘ

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), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

…

18: 17.3 $q$-Elementary and $q$-Special Functions

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17.3.1 $$e_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{x}^n}{{(q;q)}_n}=\frac{1}{{((1-q)x;q)}_{\mathrm{\infty }}},$$

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Defines:

$e_q\left(x\right)$: $q$-exponential function

Symbols:

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

Referenced by:

§17.5

Permalink:

http://dlmf.nist.gov/17.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.2 $$E_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{x}^n}{{(q;q)}_n}={(-(1-q)x;q)}_{\mathrm{\infty }}.$$

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Defines:

$E_q\left(x\right)$: $q$-exponential function

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.5

Permalink:

http://dlmf.nist.gov/17.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.3 $${\sin }_q\left(x\right)=\frac{1}{2\mathrm{i}}(e_q\left(\mathrm{i}x\right)-e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}},$$

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Defines:

${\sin }_q\left(x\right)$: $q$-sine function

Symbols:

$\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_q\left(x\right)$: $q$-exponential function, $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.4 $${\mathrm{Sin}}_q\left(x\right)=\frac{1}{2\mathrm{i}}(E_q\left(\mathrm{i}x\right)-E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{q}^{n(2n+1)}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}}.$$

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Defines:

${\mathrm{Sin}}_q\left(x\right)$: $q$-sine function

Symbols:

$\mathrm{i}$: imaginary unit, $E_q\left(x\right)$: $q$-exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.6 $${\mathrm{Cos}}_q\left(x\right)=\frac{1}{2}(E_q\left(\mathrm{i}x\right)+E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n}{q}^{n(2n-1)}{(-1)}^n{x}^{2n}}{{(q;q)}_{2n}}.$$

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Defines:

${\mathrm{Cos}}_q\left(x\right)$: $q$-cosine function

Symbols:

$\mathrm{i}$: imaginary unit, $E_q\left(x\right)$: $q$-exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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19: 8.21 Generalized Sine and Cosine Integrals

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8.21.16 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{3}{2}\right){\left(1-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}+\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k+1}\left(z\right),$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Si}(a,z)$: generalized sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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8.21.17 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{1}{2}\right){\left(\frac{1}{2}-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k}\left(z\right),$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Ci}(a,z)$: generalized cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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8.21.26 $$f(a,z)\sim z^{a-1}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(1-a\right)}_{2k}}{z^{2k}},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $f(a,z)$: auxiliary function

Permalink:

http://dlmf.nist.gov/8.21.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(viii), §8.21 and Ch.8

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8.21.27 $$g(a,z)\sim z^{a-1}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(1-a\right)}_{2k+1}}{z^{2k+1}}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $g(a,z)$: auxiliary function

Permalink:

http://dlmf.nist.gov/8.21.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(viii), §8.21 and Ch.8

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20: 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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