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

§13.6 Relations to Other Functions

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Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/13.6
Addition (effective with 1.0.14):: A subsection §13.6(vii) was added relating Kummer functions to the Coulomb functions.
See also:: Annotations for Ch.13

§13.6(i) Elementary Functions

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Keywords:: Kummer functions, elementary functions, relations to other functions
Notes:: See Temme (1996b, §7.3).
Referenced by:: §13.18(i)
Permalink:: http://dlmf.nist.gov/13.6.i
See also:: Annotations for §13.6 and Ch.13

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 Ch.13
13.6.2	$\displaystyle M\left(1,2,2z\right)$	$\displaystyle=\frac{e^{z}}{z}\sinh z,$
ⓘ 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, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 13.6.14 Permalink: http://dlmf.nist.gov/13.6.E2 Encodings: TeX, pMML, png See also: Annotations for §13.6(i), §13.6 and Ch.13
13.6.3	$\displaystyle M\left(0,b,z\right)$	$\displaystyle=U\left(0,b,z\right)=1,$
ⓘ 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, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.6.E3 Encodings: TeX, pMML, png See also: Annotations for §13.6(i), §13.6 and Ch.13
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 Ch.13

§13.6(ii) Incomplete Gamma Functions

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Keywords:: Kummer functions, confluent hypergeometric functions, error functions, incomplete gamma functions, relations to other 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).
Referenced by:: §13.18(ii)
Permalink:: http://dlmf.nist.gov/13.6.ii
See also:: Annotations for §13.6 and Ch.13

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		$M\left(a,a+1,-z\right)=e^{-z}M\left(1,a+1,z\right)=az^{-a}\gamma\left(a,z% \right),$
ⓘ 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, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable A&S Ref: 13.6.10 Permalink: http://dlmf.nist.gov/13.6.E5 Encodings: TeX, pMML, png See also: Annotations for §13.6(ii), §13.6 and Ch.13

13.6.6		$U\left(a,a,z\right)=z^{1-a}U\left(1,2-a,z\right)=z^{1-a}e^{z}E_{a}\left(z% \right)=e^{z}\Gamma\left(1-a,z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $E_{\NVar{p}}\left(\NVar{z}\right)$ : generalized exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable A&S Ref: 13.6.28 (in different form) Permalink: http://dlmf.nist.gov/13.6.E6 Encodings: TeX, pMML, png See also: Annotations for §13.6(ii), §13.6 and Ch.13

Special cases are the error functions

13.6.7		$M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)=\frac{\sqrt{\pi}}{2z}% \operatorname{erf}\left(z\right),$
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function and $z$ : 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 See also: Annotations for §13.6(ii), §13.6 and Ch.13

13.6.8		$U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)=\sqrt{\pi}e^{z^{2}}\operatorname% {erfc}\left(z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable Referenced by: §13.6(ii) Permalink: http://dlmf.nist.gov/13.6.E8 Encodings: TeX, pMML, png See also: Annotations for §13.6(ii), §13.6 and Ch.13

§13.6(iii) Modified Bessel Functions

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Keywords:: Airy functions, Kummer functions, confluent hypergeometric functions, modified Bessel functions, relations to other functions
Notes:: See Temme (1996b, §7.3.8 and p. 254).
Referenced by:: §13.18(iii)
Permalink:: http://dlmf.nist.gov/13.6.iii
Addition (effective with 1.1.0):: Equations (13.6.11_1) and (13.6.11_2) were added; they are special cases of (13.11.1) and (13.11.2), respectively.
See also:: Annotations for §13.6 and Ch.13

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	$\displaystyle M\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)$	$\displaystyle=\Gamma\left(1+\nu\right)e^{z}\left(\ifrac{z}{2}\right)^{-\nu}I_{% \nu}\left(z\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and $z$ : 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 See also: Annotations for §13.6(iii), §13.6 and Ch.13
13.6.10	$\displaystyle U\left(\nu+\tfrac{1}{2},2\nu+1,2z\right)$	$\displaystyle=\frac{1}{\sqrt{\pi}}e^{z}\left(2z\right)^{-\nu}K_{\nu}\left(z% \right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $z$ : complex variable A&S Ref: 13.6.21 Permalink: http://dlmf.nist.gov/13.6.E10 Encodings: TeX, pMML, png See also: Annotations for §13.6(iii), §13.6 and Ch.13
13.6.11	$\displaystyle U\left(\tfrac{5}{6},\tfrac{5}{3},\tfrac{4}{3}z^{3/2}\right)$	$\displaystyle=\sqrt{\pi}\frac{3^{5/6}\exp\left(\tfrac{2}{3}z^{3/2}\right)}{2^{% 2/3}z}\operatorname{Ai}\left(z\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function and $z$ : complex variable A&S Ref: 13.6.25 Referenced by: (9.10.18) Permalink: http://dlmf.nist.gov/13.6.E11 Encodings: TeX, pMML, png See also: Annotations for §13.6(iii), §13.6 and Ch.13

and in the case that $b-2a$ is an integer we have

13.6.11_1	$\displaystyle M\left(\nu+\tfrac{1}{2},2\nu+1+n,2z\right)$	$\displaystyle=\Gamma\left(\nu\right){\mathrm{e}}^{z}\left(\ifrac{z}{2}\right)^% {-\nu}\sum_{k=0}^{n}\frac{{\left(-n\right)_{k}}{\left(2\nu\right)_{k}}(\nu+k)}% {{\left(2\nu+1+n\right)_{k}}k!}I_{\nu+k}\left(z\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $n$ : nonnegative integer and $z$ : complex variable Source: Prudnikov et al. (1990, page 579, (7.11.1.6)) Referenced by: §13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.6.E11_1 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.6(iii), §13.6 and Ch.13
13.6.11_2	$\displaystyle M\left(\nu+\tfrac{1}{2},2\nu+1-n,2z\right)$	$\displaystyle=\Gamma\left(\nu-n\right){\mathrm{e}}^{z}\left(\ifrac{z}{2}\right% )^{n-\nu}\sum_{k=0}^{n}(-1)^{k}\frac{{\left(-n\right)_{k}}{\left(2\nu-2n\right% )_{k}}(\nu-n+k)}{{\left(2\nu+1-n\right)_{k}}k!}I_{\nu+k-n}\left(z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $n$ : nonnegative integer and $z$ : complex variable Source: Prudnikov et al. (1990, page 579, (7.11.1.7)) Referenced by: §13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.6.E11_2 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.6(iii), §13.6 and Ch.13

Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively

§13.6(iv) Parabolic Cylinder Functions

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Keywords:: Kummer functions, confluent hypergeometric functions, parabolic cylinder functions, relations to other functions
Notes:: See Buchholz (1969, §3.3) and Temme (1996b, §7.3.3).
Referenced by:: §13.18(iv), §13.6(v), §13.6(v)
Permalink:: http://dlmf.nist.gov/13.6.iv
See also:: Annotations for §13.6 and Ch.13

For the notation see §12.2.

13.6.12		$U\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=2^{% \frac{1}{2}a+\frac{1}{4}}e^{\frac{1}{4}z^{2}}U\left(a,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.6.E12 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

13.6.13		$U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=2^{% \frac{1}{2}a+\frac{3}{4}}\frac{e^{\frac{1}{4}z^{2}}}{z}U\left(a,z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.6.E13 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

13.6.14		$M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=\frac{% 2^{\frac{1}{2}a-\frac{3}{4}}\Gamma\left(\tfrac{1}{2}a+\tfrac{3}{4}\right)e^{% \frac{1}{4}z^{2}}}{\sqrt{\pi}}\*\left(U\left(a,z\right)+U\left(a,-z\right)% \right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.6.E14 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

13.6.15		$M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=\frac{% 2^{\frac{1}{2}a-\frac{5}{4}}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)e^{% \frac{1}{4}z^{2}}}{z\sqrt{\pi}}\*\left(U\left(a,-z\right)-U\left(a,z\right)% \right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.6.E15 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

§13.6(v) Orthogonal Polynomials

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Keywords:: Kummer functions, orthogonal polynomials, relations to confluent hypergeometric functions, relations to other functions
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 $x^{n}H_{n}\left(x\right)$ is an even function of $x$ .) For (13.6.19) see (18.11.2). For (13.6.20) combine (18.20.8) with (16.2.3), replace the ${{}_{1}F_{1}}$ notation by $M$ (§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).
Referenced by:: §13.18(v)
Permalink:: http://dlmf.nist.gov/13.6.v
See also:: Annotations for §13.6 and Ch.13

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

Hermite Polynomials

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Keywords:: Hermite polynomials, confluent hypergeometric functions, relations to other functions
See also:: Annotations for §13.6(v), §13.6 and Ch.13

13.6.16	$\displaystyle M\left(-n,\tfrac{1}{2},z^{2}\right)$	$\displaystyle=(-1)^{n}\frac{n!}{(2n)!}H_{2n}\left(z\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : 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 See also: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13
13.6.17	$\displaystyle M\left(-n,\tfrac{3}{2},z^{2}\right)$	$\displaystyle=(-1)^{n}\frac{n!}{(2n+1)!2z}H_{2n+1}\left(z\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : 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 See also: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13
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 Ch.13

Laguerre Polynomials

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Keywords:: Laguerre polynomials, relation to confluent hypergeometric functions
See also:: Annotations for §13.6(v), §13.6 and Ch.13

13.6.19		$U\left(-n,\alpha+1,z\right)=(-1)^{n}{\left(\alpha+1\right)_{n}}M\left(-n,% \alpha+1,z\right)=(-1)^{n}n!L^{(\alpha)}_{n}\left(z\right).$
ⓘ 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, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable A&S Ref: 13.6.9 and 13.6.27 Referenced by: §13.6(v), §18.17(iv), §18.9(iii) Permalink: http://dlmf.nist.gov/13.6.E19 Encodings: TeX, pMML, png See also: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

Charlier Polynomials

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Keywords:: Hahn class orthogonal polynomials, relations to confluent hypergeometric functions and generalized hypergeometric functions
See also:: Annotations for §13.6(v), §13.6 and Ch.13

13.6.20		$U\left(-n,z-n+1,a\right)={\left(-z\right)_{n}}M\left(-n,z-n+1,a\right)=a^{n}C_% {n}\left(z;a\right).$
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.6(v) Permalink: http://dlmf.nist.gov/13.6.E20 Encodings: TeX, pMML, png See also: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

§13.6(vi) Generalized Hypergeometric Functions

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Keywords:: Kummer functions, generalized hypergeometric functions, relations to other functions
Notes:: When neither $a$ nor $a-b+1$ is a nonpositive integer (13.6.21) can be verified by comparison of (13.4.17) and (16.5.1). If $a$ is a nonpositive integer, then both sides of (13.6.21) reduce to a polynomial in $z$ (compare §§13.2(i) and 16.2(iv)), and (13.6.21) follows by comparing coefficients. Similarly if $a-b+1$ is a nonpositive integer, then both sides of (13.6.21) reduce to $z^{1-b}$ times a polynomial in $z$ with identical coefficients.
Permalink:: http://dlmf.nist.gov/13.6.vi
See also:: Annotations for §13.6 and Ch.13

13.6.21		$U\left(a,b,z\right)=z^{-a}{{}_{2}F_{0}}\left(a,a-b+1;-;-z^{-1}\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable Referenced by: §13.6(vi) Permalink: http://dlmf.nist.gov/13.6.E21 Encodings: TeX, pMML, png See also: Annotations for §13.6(vi), §13.6 and Ch.13

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

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Referenced by:: §13.6, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/13.6.vii
Addition (effective with 1.0.14):: This subsection has been added.
See also:: Annotations for §13.6 and Ch.13

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

13.5 Continued Fractions 13.7 Asymptotic Expansions for Large Argument

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