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11: 8.5 Confluent Hypergeometric Representations

§8.5 Confluent Hypergeometric Representations

►For the confluent hypergeometric functions $M$, $𝐌$, $U$, and the Whittaker functions $M_{\kappa ,\mu }$ and $W_{\kappa ,\mu }$, see §§13.2(i) and 13.14(i). ►

8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

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Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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12: 15.15 Sums

§15.15 Sums

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

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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

►Here $z_0$ ($\ne 0$) is an arbitrary complex constant and the expansion converges when $|z-z_0|>\max (|z_0|,|z_0-1|)$. … ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).

13: 15.5 Derivatives and Contiguous Functions

… ►

§15.5(i) Differentiation Formulas

… ► ►

§15.5(ii) Contiguous Functions

►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. … ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …

14: 13.15 Recurrence Relations and Derivatives

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13.15.1 $$(\kappa -\mu -\frac{1}{2})M_{\kappa -1,\mu }\left(z\right)+(z-2\kappa )M_{\kappa ,\mu }\left(z\right)+(\kappa +\mu +\frac{1}{2})M_{\kappa +1,\mu }\left(z\right)=0,$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.29

Permalink:

http://dlmf.nist.gov/13.15.E1

Encodings:

pMML, png, TeX

See also:

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

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13.15.2 $$2\mu (1+2\mu )\sqrt{z}{M}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-(z+2\mu )(1+2\mu )M_{\kappa ,\mu }\left(z\right)+(\kappa +\mu +\frac{1}{2})\sqrt{z}{M}_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.15(i)

Permalink:

http://dlmf.nist.gov/13.15.E2

Encodings:

pMML, png, TeX

See also:

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

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13.15.3 $$(\kappa -\mu -\frac{1}{2})M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)+(1+2\mu )\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)-(\kappa +\mu +\frac{1}{2})M_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E3

Encodings:

pMML, png, TeX

See also:

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

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13.15.4 $$2\mu M_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-2\mu M_{\kappa +\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)=0,$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.28

Permalink:

http://dlmf.nist.gov/13.15.E4

Encodings:

pMML, png, TeX

See also:

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

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13.15.5 $$2\mu (1+2\mu )M_{\kappa ,\mu }\left(z\right)-2\mu (1+2\mu )\sqrt{z}{M}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-(\kappa -\mu -\frac{1}{2})\sqrt{z}{M}_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E5

Encodings:

pMML, png, TeX

See also:

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

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15: 13.18 Relations to Other Functions

… ►

§13.18(ii) Incomplete Gamma Functions

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§13.18(iv) Parabolic Cylinder Functions

… ►

§13.18(v) Orthogonal Polynomials

… ►

Hermite Polynomials

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Laguerre Polynomials

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16: 13.3 Recurrence Relations and Derivatives

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13.3.7 $$U(a-1,b,z)+(b-2a-z)U(a,b,z)+a(a-b+1)U(a+1,b,z)=0,$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.15

Permalink:

http://dlmf.nist.gov/13.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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13.3.8 $$(b-a-1)U(a,b-1,z)+(1-b-z)U(a,b,z)+zU(a,b+1,z)=0,$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.16

Permalink:

http://dlmf.nist.gov/13.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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13.3.9 $$U(a,b,z)-aU(a+1,b,z)-U(a,b-1,z)=0,$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.17

Permalink:

http://dlmf.nist.gov/13.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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13.3.10 $$(b-a)U(a,b,z)+U(a-1,b,z)-zU(a,b+1,z)=0,$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.18

Permalink:

http://dlmf.nist.gov/13.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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13.3.11 $$(a+z)U(a,b,z)-zU(a,b+1,z)+a(b-a-1)U(a+1,b,z)=0,$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.19

Permalink:

http://dlmf.nist.gov/13.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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17: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

§17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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Euler’s Second Sum

… ►

17.5.2 $${}_1{}^{}\varphi _0^{}(a;-;q,z)=\frac{{(az;q)}_{\mathrm{\infty }}}{{(z;q)}_{\mathrm{\infty }}},$$ ;

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Symbols:

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

Permalink:

http://dlmf.nist.gov/17.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

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Euler’s First Sum

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Cauchy’s Sum

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18: 13.14 Definitions and Basic Properties

… ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ and $W_{\kappa ,\mu }\left(z\right)$ is entire in $\kappa$ and $\mu$. Also, unless specified otherwise $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are assumed to have their principal values. … ►For $W_{\kappa ,\mu }\left(z\right)$ with use (13.14.31). …

19: 15.3 Graphics

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§15.3(i) Graphs

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Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

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Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

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Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. … Magnify 3D Help

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Figure 15.3.7: $|𝐅(-3,\frac{3}{5};u+\mathrm{i}v;\frac{1}{2})|,-6\le u\le 2,-2\le v\le 2$. Magnify 3D Help

20: 16.23 Mathematical Applications

§16.23 Mathematical Applications

… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … ►

§16.23(ii) Random Graphs

… ►

§16.23(iv) Combinatorics and Number Theory

…