# §13.6 Relations to Other Functions

## §13.6(i) Elementary Functions

 13.6.1 $\displaystyle\mathop{M\/}\nolimits\!\left(a,a,z\right)$ $\displaystyle=e^{z},$ 13.6.2 $\displaystyle\mathop{M\/}\nolimits\!\left(1,2,2z\right)$ $\displaystyle=\frac{e^{z}}{z}\mathop{\sinh\/}\nolimits z,$ 13.6.3 $\displaystyle\mathop{M\/}\nolimits\!\left(0,b,z\right)$ $\displaystyle=\mathop{U\/}\nolimits\!\left(0,b,z\right)=1,$ 13.6.4 $\displaystyle\mathop{U\/}\nolimits\!\left(a,a+1,z\right)$ $\displaystyle=z^{-a}.$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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(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 $\mathop{M\/}\nolimits\!\left(a,a+1,-z\right)=e^{-z}\mathop{M\/}\nolimits\!% \left(1,a+1,z\right)=az^{-a}\mathop{\gamma\/}\nolimits\!\left(a,z\right),$
 13.6.6 $\mathop{U\/}\nolimits\!\left(a,a,z\right)=z^{1-a}\mathop{U\/}\nolimits\!\left(% 1,2-a,z\right)=z^{1-a}e^{z}\mathop{E_{a}\/}\nolimits\!\left(z\right)=e^{z}% \mathop{\Gamma\/}\nolimits\!\left(1-a,z\right).$

Special cases are the error functions

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

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

## §13.6(iv) Parabolic Cylinder Functions

For the notation see §12.2.

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

## §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 $\displaystyle\mathop{M\/}\nolimits\!\left(-n,\tfrac{1}{2},z^{2}\right)$ $\displaystyle=(-1)^{n}\frac{n!}{(2n)!}\mathop{H_{2n}\/}\nolimits\!\left(z% \right),$ 13.6.17 $\displaystyle\mathop{M\/}\nolimits\!\left(-n,\tfrac{3}{2},z^{2}\right)$ $\displaystyle=(-1)^{n}\frac{n!}{(2n+1)!2z}\mathop{H_{2n+1}\/}\nolimits\!\left(% z\right),$ 13.6.18 $\displaystyle\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}-\tfrac{1}{2}n,\tfrac{3}% {2},z^{2}\right)$ $\displaystyle=2^{-n}z^{-1}\mathop{H_{n}\/}\nolimits\!\left(z\right).$ Symbols: $\mathop{H_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Hermite polynomial, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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)

### Laguerre Polynomials

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

### Charlier Polynomials

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

## §13.6(vi) Generalized Hypergeometric Functions

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

For the definition of $\mathop{{{}_{2}F_{0}}\/}\nolimits\!\left(a,a-b+1;-;-z^{-1}\right)$ 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).