§13.18 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/13.18

§13.18(i) Elementary Functions
§13.18(ii) Incomplete Gamma Functions
§13.18(iii) Modified Bessel Functions
§13.18(iv) Parabolic Cylinder Functions
§13.18(v) Orthogonal Polynomials

§13.18(i) Elementary Functions

Notes:: Combine §13.6(i) with (13.14.4) and (13.14.5).
Keywords:: Whittaker functions, elementary functions
Permalink:: http://dlmf.nist.gov/13.18.i

13.18.1 $\mathop{M_{{0,\frac{1}{2}}}\/}\nolimits\!\left(2z\right)=2\mathop{\sinh\/}\nolimits z,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E1
Encodings:: TeX, pMML, png

13.18.2 $\mathop{M_{{\kappa,\kappa-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\mathop{W_{{\kappa,\kappa-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\mathop{W_{{\kappa,-\kappa+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=e^{{-\frac{1}{2}z}}z^{{\kappa}},$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E2
Encodings:: TeX, pMML, png

13.18.3 $\mathop{M_{{\kappa,-\kappa-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=e^{{\frac{1}{2}z}}z^{{-\kappa}}.$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E3
Encodings:: TeX, pMML, png

§13.18(ii) Incomplete Gamma Functions

Notes:: Combine §13.6(ii) with (13.14.4) and (13.14.5).
Keywords:: Whittaker functions, confluent hypergeometric functions, error functions, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/13.18.ii

For the notation see §§6.2(i), 7.2(i), and 8.2(i). When $\tfrac{1}{2}-\kappa\pm\mu$ is an integer the Whittaker functions can be expressed as incomplete gamma functions (or generalized exponential integrals). For example,

13.18.4 $\mathop{M_{{\mu-\frac{1}{2},\mu}}\/}\nolimits\!\left(z\right)=2\mu e^{{\frac{1}{2}z}}z^{{\frac{1}{2}-\mu}}\mathop{\gamma\/}\nolimits\!\left(2\mu,z\right),$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $z$ : complex variable
Referenced by:: §8.5
Permalink:: http://dlmf.nist.gov/13.18.E4
Encodings:: TeX, pMML, png

13.18.5 $\mathop{W_{{\mu-\frac{1}{2},\mu}}\/}\nolimits\!\left(z\right)=e^{{\frac{1}{2}z}}z^{{\frac{1}{2}-\mu}}\mathop{\Gamma\/}\nolimits\!\left(2\mu,z\right).$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E5
Encodings:: TeX, pMML, png

Special cases are the error functions

13.18.6 $\mathop{M_{{-\frac{1}{4},\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)=\tfrac{1}{2}e^{{\frac{1}{2}z^{2}}}\sqrt{\pi z}\mathop{\mathrm{erf}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E6
Encodings:: TeX, pMML, png

13.18.7 $\mathop{W_{{-\frac{1}{4},\pm\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)=e^{{\frac{1}{2}z^{2}}}\sqrt{\pi z}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function and $z$ : complex variable
Referenced by:: Version 1.0.3 (Aug 29, 2011)
Permalink:: http://dlmf.nist.gov/13.18.E7
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.3):: Originally the left-hand side was given correctly as $\mathop{W_{{-\frac{1}{4},-\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)$ ; the equation is true also for $\mathop{W_{{-\frac{1}{4},+\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)$ .
Reported 2011-08-21

§13.18(iii) Modified Bessel Functions

Notes:: Combine §13.6(iii) with (13.14.4) and (13.14.5).
Keywords:: Airy functions, Whittaker functions, confluent hypergeometric functions, modified Bessel functions
Permalink:: http://dlmf.nist.gov/13.18.iii

When $\kappa=0$ the Whittaker functions can be expressed as modified Bessel functions. For the notation see §§10.25(ii) and 9.2(i).

13.18.8 $\mathop{M_{{0,\nu}}\/}\nolimits\!\left(2z\right)=2^{{2\nu+\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(1+\nu\right)\sqrt{z}\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $z$ : complex variable
Referenced by:: §13.24(ii)
Permalink:: http://dlmf.nist.gov/13.18.E8
Encodings:: TeX, pMML, png

13.18.9 $\mathop{W_{{0,\nu}}\/}\nolimits\!\left(2z\right)=\sqrt{\ifrac{2z}{\pi}}\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E9
Encodings:: TeX, pMML, png

13.18.10 $\mathop{W_{{0,\frac{1}{3}}}\/}\nolimits\!\left(\tfrac{4}{3}z^{{\frac{3}{2}}}\right)=2\sqrt{\pi}z^{{\frac{1}{4}}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E10
Encodings:: TeX, pMML, png

§13.18(iv) Parabolic Cylinder Functions

Notes:: Combine §13.6(iv) with (13.14.4) and (13.14.5).
Keywords:: Whittaker functions, confluent hypergeometric functions, parabolic cylinder functions
Referenced by:: §13.18(v)
Permalink:: http://dlmf.nist.gov/13.18.iv

For the notation see §12.2.

13.18.11 $\mathop{W_{{-\frac{1}{2}a,\pm\frac{1}{4}}}\/}\nolimits\!\left(\tfrac{1}{2}z^{{2}}\right)=2^{{\frac{1}{2}a}}\sqrt{z}\mathop{U\/}\nolimits(a,z),$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E11
Encodings:: TeX, pMML, png

13.18.12 $\mathop{M_{{-\frac{1}{2}a,-\frac{1}{4}}}\/}\nolimits\!\left(\tfrac{1}{2}z^{{2}}\right)=2^{{\frac{1}{2}a-1}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{3}{4}\right)\sqrt{\ifrac{z}{\pi}}\*\left(\mathop{U\/}\nolimits\!\left(a,z\right)+\mathop{U\/}\nolimits\!\left(a,-z\right)\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E12
Encodings:: TeX, pMML, png

13.18.13 $\mathop{M_{{-\frac{1}{2}a,\frac{1}{4}}}\/}\nolimits\!\left(\tfrac{1}{2}z^{{2}}\right)=2^{{\frac{1}{2}a-2}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\sqrt{\ifrac{z}{\pi}}\*\left(\mathop{U\/}\nolimits\!\left(a,-z\right)-\mathop{U\/}\nolimits\!\left(a,z\right)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E13
Encodings:: TeX, pMML, png

§13.18(v) Orthogonal Polynomials

Notes:: Combine §13.6(v) with (13.14.4) and (13.14.5).
Keywords:: Whittaker functions, orthogonal polynomials
Permalink:: http://dlmf.nist.gov/13.18.v

Special cases of §13.18(iv) are as follows. For the notation see §18.3.

¶ Hermite Polynomials

Keywords:: Hermite polynomials, confluent hypergeometric functions

13.18.14 $\mathop{M_{{\frac{1}{4}+n,-\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)=(-1)^{n}\frac{n!}{(2n)!}e^{{-\frac{1}{2}z^{2}}}\sqrt{z}\mathop{H_{{2n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E14
Encodings:: TeX, pMML, png

13.18.15 $\mathop{M_{{\frac{3}{4}+n,\frac{1}{4}}}\/}\nolimits\!\left(z^{2}\right)=(-1)^{n}\frac{n!}{(2n+1)!}\frac{e^{{-\frac{1}{2}z^{2}}}\sqrt{z}}{2}\mathop{H_{{2n+1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E15
Encodings:: TeX, pMML, png

13.18.16 $\mathop{W_{{\frac{1}{4}+\frac{1}{2}n,\frac{1}{4}}}\/}\nolimits\!\left(z^{{2}}\right)=2^{{-n}}e^{{-\frac{1}{2}z^{2}}}\sqrt{z}\mathop{H_{{n}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E16
Encodings:: TeX, pMML, png

¶ Laguerre Polynomials

Keywords:: Laguerre polynomials

13.18.17 $\mathop{W_{{\frac{1}{2}\alpha+\frac{1}{2}+n,\frac{1}{2}\alpha}}\/}\nolimits\!\left(z\right)=(-1)^{{n}}\left(\alpha+1\right)_{{n}}\mathop{M_{{\frac{1}{2}\alpha+\frac{1}{2}+n,\frac{1}{2}\alpha}}\/}\nolimits\!\left(z\right)=(-1)^{{n}}n!e^{{-\frac{1}{2}z}}z^{{\frac{1}{2}\alpha+\frac{1}{2}}}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(z\right).$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.18.E17
Encodings:: TeX, pMML, png