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31: 13.6 Relations to Other Functions

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§13.6(v) Orthogonal Polynomials

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Hermite Polynomials

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Laguerre Polynomials

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Charlier Polynomials

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§13.6(vi) Generalized Hypergeometric Functions

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32: 13.18 Relations to Other Functions

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§13.18(ii) Incomplete Gamma Functions

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§13.18(iv) Parabolic Cylinder Functions

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§13.18(v) Orthogonal Polynomials

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Hermite Polynomials

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Laguerre Polynomials

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33: 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). … ►

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

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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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8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

34: 15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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Jacobi

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Legendre

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Meixner

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§15.9(ii) Jacobi Function

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35: 16.6 Transformations of Variable

§16.6 Transformations of Variable

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Quadratic

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Cubic

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16.6.2 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a,2b-a-1,2-2b+a}{b,a-b+\frac{3}{2}};\frac{z}{4})={(1-z)}^{-a}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}}{b,a-b+\frac{3}{2}};\frac{-27z}{4{(1-z)}^3}).$$

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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, $z$: 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

Referenced by:

§16.6

Permalink:

http://dlmf.nist.gov/16.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.6, §16.6 and Ch.16

►For Kummer-type transformations of ${}_2{}^{}F_2^{}$ functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).

36: 16.4 Argument Unity

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Watson’s Sum

… ► … ► … ►Denote, formally, the bilateral hypergeometric function …

37: 10.39 Relations to Other Functions

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Confluent Hypergeometric Functions

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10.39.7 $$I_{\nu }\left(z\right)=\frac{{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(2z\right)}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.47

Referenced by:

(9.6.23), (9.6.24)

Permalink:

http://dlmf.nist.gov/10.39.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

… ►For the functions $M$, $U$, $M_{0,\nu }$, and $W_{0,\nu }$ see §§13.2(i) and 13.14(i). ►

Generalized Hypergeometric Functions and Hypergeometric Function

… ►For the functions ${}_0{}^{}F_1^{}$ and $𝐅$ see (16.2.1) and §15.2(i).

38: 12.18 Methods of Computation

… ►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. …

39: 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

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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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40: 13.2 Definitions and Basic Properties

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13.2.4 $$M(a,b,z)=\mathrm{\Gamma }\left(b\right)𝐌(a,b,z).$$

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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 and $z$: complex variable

Referenced by:

§13.2(ii)

Permalink:

http://dlmf.nist.gov/13.2.E4

Encodings:

pMML, png, TeX

See also:

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

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13.2.34 $$𝒲\left\{𝐌(a,b,z),U(a,b,z)\right\}=-z^{-b}{\mathrm{e}}^z/\mathrm{\Gamma }\left(a\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $𝒲$: Wronskian, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.1.22

Permalink:

http://dlmf.nist.gov/13.2.E34

Encodings:

pMML, png, TeX

See also:

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

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13.2.35 $$𝒲\left\{𝐌(a,b,z),{\mathrm{e}}^{z}U(b-a,b,{\mathrm{e}}^{\pm \pi \mathrm{i}}z)\right\}={\mathrm{e}}^{\mp b\pi \mathrm{i}}{z}^{-b}{\mathrm{e}}^z/\mathrm{\Gamma }\left(b-a\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

13.1.23

Permalink:

http://dlmf.nist.gov/13.2.E35

Encodings:

pMML, png, TeX

See also:

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

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13.2.36 $$𝒲\left\{z^{1-b}𝐌(a-b+1,2-b,z),U(a,b,z)\right\}=-z^{-b}{\mathrm{e}}^z/\mathrm{\Gamma }\left(a-b+1\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $𝒲$: Wronskian, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.1.24

Permalink:

http://dlmf.nist.gov/13.2.E36

Encodings:

pMML, png, TeX

See also:

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

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Kummer’s Transformations

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