13 Confluent Hypergeometric Functions Kummer Functions 13.5 Continued Fractions 13.7 Asymptotic Expansions for Large Argument

§13.6 Relations to Other Functions

Referenced by:: ¶ ‣ §3.10(ii)
Permalink:: http://dlmf.nist.gov/13.6

§13.6(i) Elementary Functions
§13.6(ii) Incomplete Gamma Functions
§13.6(iii) Modified Bessel Functions
§13.6(iv) Parabolic Cylinder Functions
§13.6(v) Orthogonal Polynomials
§13.6(vi) Generalized Hypergeometric Functions

§13.6(i) Elementary Functions

Notes:: See Temme (1996b, §7.3).
Keywords:: Kummer functions, elementary functions
Referenced by:: §13.18(i)
Permalink:: http://dlmf.nist.gov/13.6.i

13.6.1 $\mathop{M\/}\nolimits\!\left(a,a,z\right)=e^{{z}},$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function and : 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

13.6.2 $\mathop{M\/}\nolimits\!\left(1,2,2z\right)=\frac{e^{{z}}}{z}\mathop{\sinh\/}% \nolimits z,$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 13.6.14
Permalink:: http://dlmf.nist.gov/13.6.E2
Encodings:: TeX, pMML, png

13.6.3 $\mathop{M\/}\nolimits\!\left(0,b,z\right)=\mathop{U\/}\nolimits\!\left(0,b,z% \right)=1,$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.6.E3
Encodings:: TeX, pMML, png

13.6.4 $\mathop{U\/}\nolimits\!\left(a,a+1,z\right)=z^{{-a}}.$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.6.E4
Encodings:: TeX, pMML, png

§13.6(ii) Incomplete Gamma Functions

Notes:: See Temme (1996b, §7.3: each of the equations on p. 180 that correspond to (13.6.7) and (13.6.8) contains an error).
Keywords:: Kummer functions, confluent hypergeometric functions, error functions, incomplete gamma functions
Referenced by:: §13.18(ii)
Permalink:: http://dlmf.nist.gov/13.6.ii

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

13.6.5 $\mathop{M\/}\nolimits\!\left(a,a+1,-z\right)=e^{{-z}}\mathop{M\/}\nolimits\!% \left(1,a+1,z\right)=az^{{-a}}\mathop{\gamma\/}\nolimits\!\left(a,z\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : complex variable
A&S Ref:: 13.6.10
Permalink:: http://dlmf.nist.gov/13.6.E5
Encodings:: TeX, pMML, png

13.6.6 $\mathop{U\/}\nolimits\!\left(a,a,z\right)=z^{{1-a}}\mathop{U\/}\nolimits\!% \left(1,2-a,z\right)=z^{{1-a}}e^{{z}}\mathop{E_{{a}}\/}\nolimits\!\left(z% \right)=e^{{z}}\mathop{\Gamma\/}\nolimits\!\left(1-a,z\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ : generalized exponential integral, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : complex variable
A&S Ref:: 13.6.28 (in different form)
Permalink:: http://dlmf.nist.gov/13.6.E6
Encodings:: TeX, pMML, png

Special cases are the error functions

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

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\mathrm{erf}\/}\nolimits z$ : error function and : complex variable
A&S Ref:: 13.6.19
Referenced by:: §13.6(ii)
Permalink:: http://dlmf.nist.gov/13.6.E7
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, : base of exponential function and : complex variable
Referenced by:: §13.6(ii)
Permalink:: http://dlmf.nist.gov/13.6.E8
Encodings:: TeX, pMML, png

§13.6(iii) Modified Bessel Functions

Notes:: See Temme (1996b, §7.3.8 and p. 254).
Keywords:: Airy functions, Kummer functions, confluent hypergeometric functions, modified Bessel functions
Referenced by:: §13.18(iii)
Permalink:: http://dlmf.nist.gov/13.6.iii

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

13.6.9 $\mathop{M\/}\nolimits\!\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)=\mathop{\Gamma% \/}\nolimits\!\left(1+\nu\right)e^{{z}}\left(\ifrac{z}{2}\right)^{{-\nu}}% \mathop{I_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : complex variable
A&S Ref:: 13.6.3
Referenced by:: §13.11, §7.6(ii)
Permalink:: http://dlmf.nist.gov/13.6.E9
Encodings:: TeX, pMML, png

13.6.10 $\mathop{U\/}\nolimits\!\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)=\frac{1}{\sqrt{% \pi}}e^{{z}}\left(2z\right)^{{-\nu}}\mathop{K_{{\nu}}\/}\nolimits\!\left(z% \right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : complex variable
A&S Ref:: 13.6.21
Permalink:: http://dlmf.nist.gov/13.6.E10
Encodings:: TeX, pMML, png

13.6.11 $\mathop{U\/}\nolimits\!\left(\tfrac{5}{6},\tfrac{5}{3},\tfrac{4}{3}z^{{3/2}}% \right)=\sqrt{\pi}\frac{3^{{5/6}}\mathop{\exp\/}\nolimits\!\left(\tfrac{2}{3}z% ^{{3/2}}\right)}{2^{{2/3}}z}\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\exp\/}\nolimits z$ : exponential function and : complex variable
A&S Ref:: 13.6.25
Permalink:: http://dlmf.nist.gov/13.6.E11
Encodings:: TeX, pMML, png

§13.6(iv) Parabolic Cylinder Functions

Notes:: See Buchholz (1969, §3.3) and Temme (1996b, §7.3.3).
Keywords:: Kummer functions, confluent hypergeometric functions, parabolic cylinder functions
Referenced by:: §13.18(iv), §13.6(v), §13.6(v)
Permalink:: http://dlmf.nist.gov/13.6.iv

For the notation see §12.2.

13.6.12 $\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}% {2}z^{{2}}\right)=2^{{\frac{1}{2}a+\frac{1}{4}}}e^{{\frac{1}{4}z^{2}}}\mathop{% U\/}\nolimits\!\left(a,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and : complex variable
Permalink:: http://dlmf.nist.gov/13.6.E12
Encodings:: TeX, pMML, png

13.6.13 $\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}% {2}z^{{2}}\right)=2^{{\frac{1}{2}a+\frac{3}{4}}}\frac{e^{{\frac{1}{4}z^{2}}}}{% z}\mathop{U\/}\nolimits\!\left(a,z\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and : complex variable
Permalink:: http://dlmf.nist.gov/13.6.E13
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and : complex variable
Permalink:: http://dlmf.nist.gov/13.6.E14
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function and : complex variable
Permalink:: http://dlmf.nist.gov/13.6.E15
Encodings:: TeX, pMML, png

§13.6(v) Orthogonal Polynomials

Notes:: For (13.6.16)–(13.6.18) combine §13.6(iv) with §12.7(i) and (18.5.18). (The last equation is needed to illustrate that $x^{n}\!\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ is an even function of .) For (13.6.19) see (18.11.2). For (13.6.20) combine (18.20.8) with (16.2.3), replace the $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits$ notation by $\mathop{M\/}\nolimits$ (§13.1), and then use (13.2.42) (in which the final term vanishes). Alternatively, for (13.6.20) combine (16.2.3) with Andrews et al. (1999, p. 347).
Keywords:: Kummer functions, orthogonal polynomials
Referenced by:: §13.18(v)
Permalink:: http://dlmf.nist.gov/13.6.v

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

¶ Hermite Polynomials

Keywords:: Hermite polynomials, confluent hypergeometric functions

13.6.16 $\mathop{M\/}\nolimits\!\left(-n,\tfrac{1}{2},z^{{2}}\right)=(-1)^{n}\frac{n!}{% (2n)!}\mathop{H_{{2n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : : factorial, : nonnegative integer and : complex variable
A&S Ref:: 13.6.17
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/13.6.E16
Encodings:: TeX, pMML, png

13.6.17 $\mathop{M\/}\nolimits\!\left(-n,\tfrac{3}{2},z^{{2}}\right)=(-1)^{n}\frac{n!}{% (2n+1)!2z}\mathop{H_{{2n+1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : : factorial, : nonnegative integer and : complex variable
A&S Ref:: 13.6.18 (as corrected in later editions)
Permalink:: http://dlmf.nist.gov/13.6.E17
Encodings:: TeX, pMML, png

13.6.18 $\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}-\tfrac{1}{2}n,\tfrac{3}{2},z^{{2}}% \right)=2^{{-n}}z^{{-1}}\mathop{H_{{n}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : nonnegative integer and : 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

¶ Laguerre Polynomials

Keywords:: Laguerre polynomials

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

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : nonnegative integer and : complex variable
A&S Ref:: 13.6.9 and 13.6.27
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/13.6.E19
Encodings:: TeX, pMML, png

¶ Charlier Polynomials

Keywords:: Hahn class orthogonal polynomials

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

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : nonnegative integer and : complex variable
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/13.6.E20
Encodings:: TeX, pMML, png

§13.6(vi) Generalized Hypergeometric Functions

Notes:: When neither nor is a nonpositive integer (13.6.21) can be verified by comparison of (13.4.17) and (16.5.1). If is a nonpositive integer, then both sides of (13.6.21) reduce to a polynomial in (compare §§13.2(i) and 16.2(iv)), and (13.6.21) follows by comparing coefficients. Similarly if is a nonpositive integer, then both sides of (13.6.21) reduce to $z^{{1-b}}$ times a polynomial in with identical coefficients.
Keywords:: Kummer functions, generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/13.6.vi