§13.6 Relations to Other Functions

§13.6(i) Elementary Functions

 13.6.1 $\displaystyle M\left(a,a,z\right)$ $\displaystyle=e^{z},$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 13.6.12 Referenced by: §13.10(v) Permalink: http://dlmf.nist.gov/13.6.E1 Encodings: TeX, pMML, png See also: Annotations for 13.6(i), 13.6 and 13 13.6.2 $\displaystyle M\left(1,2,2z\right)$ $\displaystyle=\frac{e^{z}}{z}\sinh z,$ 13.6.3 $\displaystyle M\left(0,b,z\right)$ $\displaystyle=U\left(0,b,z\right)=1,$ 13.6.4 $\displaystyle U\left(a,a+1,z\right)$ $\displaystyle=z^{-a}.$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.6.E4 Encodings: TeX, pMML, png See also: Annotations for 13.6(i), 13.6 and 13

§13.6(ii) Incomplete Gamma Functions

For the notation see §§6.2(i), 7.2(i), 8.2(i), and 8.19(i). When $a-b$ is an integer or $a$ is a positive integer the Kummer functions can be expressed as incomplete gamma functions (or generalized exponential integrals). For example,

 13.6.5 $M\left(a,a+1,-z\right)=e^{-z}M\left(1,a+1,z\right)=az^{-a}\gamma\left(a,z% \right),$
 13.6.6 $U\left(a,a,z\right)=z^{1-a}U\left(1,2-a,z\right)=z^{1-a}e^{z}E_{a}\left(z% \right)=e^{z}\Gamma\left(1-a,z\right).$

Special cases are the error functions

 13.6.7 $M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)=\frac{\sqrt{\pi}}{2z}% \operatorname{erf}\left(z\right),$
 13.6.8 $U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)=\sqrt{\pi}e^{z^{2}}\operatorname% {erfc}\left(z\right).$

§13.6(iii) Modified Bessel Functions

When $b=2a$ the Kummer functions can be expressed as modified Bessel functions. For the notation see §§10.25(ii) and 9.2(i).

 13.6.9 $\displaystyle M\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)$ $\displaystyle=\Gamma\left(1+\nu\right)e^{z}\left(\ifrac{z}{2}\right)^{-\nu}I_{% \nu}\left(z\right),$ 13.6.10 $\displaystyle U\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)$ $\displaystyle=\frac{1}{\sqrt{\pi}}e^{z}\left(2z\right)^{-\nu}K_{\nu}\left(z% \right),$ 13.6.11 $\displaystyle U\left(\tfrac{5}{6},\tfrac{5}{3},\tfrac{4}{3}z^{3/2}\right)$ $\displaystyle=\sqrt{\pi}\frac{3^{5/6}\exp\left(\tfrac{2}{3}z^{3/2}\right)}{2^{% 2/3}z}\mathrm{Ai}\left(z\right).$

§13.6(iv) Parabolic Cylinder Functions

For the notation see §12.2.

 13.6.12 $U\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=2^{% \frac{1}{2}a+\frac{1}{4}}e^{\frac{1}{4}z^{2}}U\left(a,z\right),$
 13.6.13 $U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=2^{% \frac{1}{2}a+\frac{3}{4}}\frac{e^{\frac{1}{4}z^{2}}}{z}U\left(a,z\right).$
 13.6.14 $M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=\frac{% 2^{\frac{1}{2}a-\frac{3}{4}}\Gamma\left(\tfrac{1}{2}a+\tfrac{3}{4}\right)e^{% \frac{1}{4}z^{2}}}{\sqrt{\pi}}\*\left(U\left(a,z\right)+U\left(a,-z\right)% \right),$
 13.6.15 $M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=\frac{% 2^{\frac{1}{2}a-\frac{5}{4}}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)e^{% \frac{1}{4}z^{2}}}{z\sqrt{\pi}}\*\left(U\left(a,-z\right)-U\left(a,z\right)% \right).$

§13.6(v) Orthogonal Polynomials

Special cases of §13.6(iv) are as follows. For the notation see §§18.3, 18.19.

Hermite Polynomials

 13.6.16 $\displaystyle M\left(-n,\tfrac{1}{2},z^{2}\right)$ $\displaystyle=(-1)^{n}\frac{n!}{(2n)!}H_{2n}\left(z\right),$ 13.6.17 $\displaystyle M\left(-n,\tfrac{3}{2},z^{2}\right)$ $\displaystyle=(-1)^{n}\frac{n!}{(2n+1)!2z}H_{2n+1}\left(z\right),$ 13.6.18 $\displaystyle U\left(\tfrac{1}{2}-\tfrac{1}{2}n,\tfrac{3}{2},z^{2}\right)$ $\displaystyle=2^{-n}z^{-1}H_{n}\left(z\right).$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable A&S Ref: 13.6.38 (as corrected in later editions) Referenced by: §13.6(v) Permalink: http://dlmf.nist.gov/13.6.E18 Encodings: TeX, pMML, png See also: Annotations for 13.6(v), 13.6(v), 13.6 and 13

Laguerre Polynomials

 13.6.19 $U\left(-n,\alpha+1,z\right)=(-1)^{n}{\left(\alpha+1\right)_{n}}M\left(-n,% \alpha+1,z\right)=(-1)^{n}n!L^{(\alpha)}_{n}\left(z\right).$

Charlier Polynomials

 13.6.20 $U\left(-n,z-n+1,a\right)={\left(-z\right)_{n}}M\left(-n,z-n+1,a\right)=a^{n}C_% {n}\left(z;a\right).$

§13.6(vi) Generalized Hypergeometric Functions

 13.6.21 $U\left(a,b,z\right)=z^{-a}{{}_{2}F_{0}}\left(a,a-b+1;-;-z^{-1}\right).$

For the definition of ${{}_{2}F_{0}}\left(a,a-b+1;-;-z^{-1}\right)$ when neither $a$ nor $a-b+1$ is a nonpositive integer see §16.5.

§13.6(vii) Coulomb Functions

For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).