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21: 15.15 Sums

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15.15.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};\frac{1}{z})={\left(1-\frac{z_0}{z}\right)}^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{s!}𝐅(\genfrac{}{}{0pt}{}{-s,b}{c};\frac{1}{z_0}){\left(1-\frac{z}{z_0}\right)}^{-s}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.19(i)

Permalink:

http://dlmf.nist.gov/15.15.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §15.15 and Ch.15

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22: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

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28.19.1 $$f(z+\pi )={\mathrm{e}}^{\mathrm{i}\nu \pi }f(z).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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28.19.2 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{\mathrm{me}}_{\nu +2n}(z,q),$$

ⓘ

Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f_n$: coefficients

Referenced by:

§28.19

Permalink:

http://dlmf.nist.gov/28.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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28.19.3 $$f_n=\frac{1}{\pi }{\int }_0^{\pi }f(z){\mathrm{me}}_{\nu +2n}(-z,q)dz.$$

ⓘ

Defines:

$f_n$: coefficients (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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28.19.4 $${\mathrm{e}}^{\mathrm{i}\nu z}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{-2n}^{\nu +2n}(q){\mathrm{me}}_{\nu +2n}(z,q),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19, §28.19 and Ch.28

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23: 15.10 Hypergeometric Differential Equation

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15.10.1 $$z(1-z)\frac{d^{2}w}{{dz}^2}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)

Permalink:

http://dlmf.nist.gov/15.10.E1

Encodings:

pMML, png, TeX

See also:

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

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15.10.3 $$𝒲\left\{f_1(z),f_2(z)\right\}=(1-c)z^{-c}{(1-z)}^{c-a-b-1}.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E3

Encodings:

pMML, png, TeX

See also:

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

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15.10.5 $$𝒲\left\{f_1(z),f_2(z)\right\}=(a+b-c)z^{-c}{(1-z)}^{c-a-b-1}.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E5

Encodings:

pMML, png, TeX

See also:

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

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15.10.7 $$𝒲\left\{f_1(z),f_2(z)\right\}=(a-b)z^{-c}{(z-1)}^{c-a-b-1}.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E7

Encodings:

pMML, png, TeX

See also:

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

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15.10.21 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_3(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a+b-c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_4(z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_3(z)$: Kummer’s solution $w_3$ and $w_4(z)$: Kummer’s solution $w_4$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E21

Encodings:

pMML, png, TeX

See also:

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

…

24: 12.8 Recurrence Relations and Derivatives

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12.8.1 $$zU(a,z)-U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.4

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.2 $$U^{\prime }(a,z)+\frac{1}{2}zU(a,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.1

Permalink:

http://dlmf.nist.gov/12.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.3 $$U^{\prime }(a,z)-\frac{1}{2}zU(a,z)+U(a-1,z)=0,$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.2

Permalink:

http://dlmf.nist.gov/12.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

►

12.8.4 $$2U^{\prime }(a,z)+U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0.$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.3

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.5 $$zV(a,z)-V(a+1,z)+(a-\frac{1}{2})V(a-1,z)=0,$$

ⓘ

Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.8

Permalink:

http://dlmf.nist.gov/12.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

…

25: 31.8 Solutions via Quadratures

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31.8.2 $$w_{\pm }(𝐦;\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.8 and Ch.31

…

26: 31.17 Physical Applications

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31.17.2 $$\frac{x_s^2}{z_k}+\frac{x_t^2}{z_k-1}+\frac{x_u^2}{z_k-a}=0,$$ $k=1,2$,

ⓘ

Symbols:

$z$: complex variable, $a$: complex parameter and $x_s$, $x_t$, $x_u$: coordinates

Permalink:

http://dlmf.nist.gov/31.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

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31.17.4 $$\mathrm{\Psi }(𝐱)={(z_1{z}_2)}^{-s-\frac{1}{4}}{((z_1-1)(z_2-1))}^{-t-\frac{1}{4}}{((z_1-a)(z_2-a))}^{-u-\frac{1}{4}}w(z_1)w(z_2),$$

ⓘ

Symbols:

$z$: complex variable, $a$: complex parameter and $\mathrm{\Psi }(𝐱)$: common eigenfunction

Permalink:

http://dlmf.nist.gov/31.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

…

27: 12.4 Power-Series Expansions

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12.4.1 $$U(a,z)=U(a,0)u_1(a,z)+U^{\prime }(a,0)u_2(a,z),$$

ⓘ

Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.2 $$V(a,z)=V(a,0)u_1(a,z)+V^{\prime }(a,0)u_2(a,z),$$

ⓘ

Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.5 $$u_1(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(1+(a-\frac{1}{2})\frac{z^2}{2!}+(a-\frac{1}{2})(a-\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_1(a,z)$: solution

A&S Ref:

19.2.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.6 $$u_2(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(z+(a-\frac{3}{2})\frac{z^3}{3!}+(a-\frac{3}{2})(a-\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_2(a,z)$: solution

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

…

28: 10.13 Other Differential Equations

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10.13.1 $$w^{\prime \prime }+\left({\lambda }^2-\frac{{\nu }^2-\frac{1}{4}}{z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.49

Referenced by:

§1.18(vi), §10.36

Permalink:

http://dlmf.nist.gov/10.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.2 $$w^{\prime \prime }+\left(\frac{{\lambda }^2}{4z}-\frac{{\nu }^2-1}{4z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z^{\frac{1}{2}}\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.50

Permalink:

http://dlmf.nist.gov/10.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.5 $$z^2{w}^{\prime \prime }+(1-2r)z{w}^{\prime }+({\lambda }^2{q}^2{z}^{2q}+r^2-{\nu }^2{q}^2)w=0,$$ $w=z^r{𝒞}_{\nu }\left(\lambda z^q\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.53

Permalink:

http://dlmf.nist.gov/10.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.9 $$z^2{w}^{\prime \prime \prime }+3z{w}^{\prime \prime }+(4z^2+1-4{\nu }^2)w^{\prime }+4zw=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.58 (modified)

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.10 $$z^3{w}^{\prime \prime \prime }+z(4z^2+1-4{\nu }^2)w^{\prime }+(4{\nu }^2-1)w=0,$$ $w=z{𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.59

Permalink:

http://dlmf.nist.gov/10.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

…

29: 16.4 Argument Unity

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16.4.1 $$a_1+b_1=\mathrm{\cdots }=a_q+b_q=a_{q+1}+1.$$

ⓘ

Symbols:

$q$: nonnegative integer, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.4(i), §16.4 and Ch.16

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16.4.2 $$a_1+\mathrm{\cdots }+a_{q+1}+k=b_1+\mathrm{\cdots }+b_q.$$

ⓘ

Symbols:

$q$: nonnegative integer, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.4(i), §16.4 and Ch.16

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16.4.11 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};1)=\frac{\mathrm{\Gamma }\left(e\right)\mathrm{\Gamma }\left(d+e-a-b-c\right)}{\mathrm{\Gamma }\left(e-a\right)\mathrm{\Gamma }\left(d+e-b-c\right)}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,d-b,d-c}{d,d+e-b-c};1),$$

ⓘ

Symbols:

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

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/16.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §16.4(iii), §16.4 and Ch.16

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16.4.12 $$(a-d)(b-d)(c-d)\left({}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d+1,e};1)-{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};1)\right)+abc{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};1)=d(d-1)(a+b+c-d-e+1)\left({}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};1)-{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d-1,e};1)\right),$$

ⓘ

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, $a$: real or complex parameters, $b$: real or complex parameters, $c$: real or complex parameters, $d$: real or complex parameters and $e$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §16.4(iii), §16.4 and Ch.16

… ►

16.4.13 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};1)=\frac{c(e-a)}{de}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b+1,c+1}{d+1,e+1};1)+\frac{d-c}{d}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,b+1,c}{d+1,e};1).$$

ⓘ

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, $a$: real or complex parameters, $b$: real or complex parameters, $c$: real or complex parameters, $d$: real or complex parameters and $e$: real or complex parameters

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/16.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §16.4(iii), §16.4 and Ch.16

…

30: 15.2 Definitions and Analytical Properties

… ►

15.2.2 $$𝐅(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$ ,

ⓘ

Defines:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §16.2(v)

Permalink:

http://dlmf.nist.gov/15.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

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15.2.3 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};x+\mathrm{i}0)-𝐅(\genfrac{}{}{0pt}{}{a,b}{c};x-\mathrm{i}0)=\frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{(x-1)}^{c-a-b}𝐅(\genfrac{}{}{0pt}{}{c-a,c-b}{c-a-b+1};1-x),$$ $x>1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.2(i)

Permalink:

http://dlmf.nist.gov/15.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

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15.2.4 $$F(-m,b;c;z)=\sum _{n=0}^m\frac{{\left(-m\right)}_n{\left(b\right)}_n}{{\left(c\right)}_{n}n!}{z}^n=\sum _{n=0}^m{(-1)}^n\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)\frac{{\left(b\right)}_n}{{\left(c\right)}_n}{z}^n.$$

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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), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $n$: integer, $z$: complex variable, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

A&S Ref:

15.4.1

Permalink:

http://dlmf.nist.gov/15.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

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15.2.5 $$F(\genfrac{}{}{0pt}{}{-m,b}{-m-\mathrm{\ell }};z)=\underset{c\to -m-\mathrm{\ell }}{lim}\left(\underset{a\to -m}{lim}F(\genfrac{}{}{0pt}{}{a,b}{c};z)\right),$$

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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, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\mathrm{\ell }$: integer and $m$: integer

Referenced by:

§15.10(i), §15.2(ii), §15.2(ii), §15.4(i), §15.4(iii)

Permalink:

http://dlmf.nist.gov/15.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

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15.2.6 $$F(\genfrac{}{}{0pt}{}{-m,b}{-m-\mathrm{\ell }};z)=\underset{a\to -m}{lim}F(\genfrac{}{}{0pt}{}{a,b}{a-\mathrm{\ell }};z),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $\mathrm{\ell }$: integer and $m$: integer

Referenced by:

§15.2(ii), §15.2(ii), §15.4(i), §15.4(iii), Erratum (V1.0.16) for Subsection 15.2(ii)

Permalink:

http://dlmf.nist.gov/15.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

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