13 Confluent Hypergeometric Functions Kummer Functions 13.10 Integrals 13.12 Products

§13.11 Series

ⓘ

Keywords:: Kummer functions, in modified Bessel functions, series expansions
Notes:: See Slater (1960, §2.7.3: Eq. (2.7.14) has errors).
Referenced by:: Erratum (V1.0.11) for References, Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/13.11
Addition (effective with 1.1.0):: Equations (13.11.2), (13.11.3) and (13.11.4) were added.
Addition (effective with 1.0.11):: The reference to López and Pérez Sinusía (2014) was added at the end of this section.
Addition (effective with 1.0.5):: The reference to López and Temme (2010a) has been added at the end of this subsection.
See also:: Annotations for Ch.13

For $z\in\mathbb{C}$ ,

13.11.1		$M\left(a,b,z\right)=\Gamma\left(a-\tfrac{1}{2}\right)e^{\frac{1}{2}z}\left(% \tfrac{1}{4}z\right)^{\frac{1}{2}-a}\\sum_{s=0}^{\infty}\frac{{\left(2a-1% \right)_{s}}{\left(2a-b\right)_{s}}}{{\left(b\right)_{s}}s!}\\left(a-\tfrac{1% }{2}+s\right)\*I_{a-\frac{1}{2}+s}\left(\tfrac{1}{2}z\right),$
$a+\frac{1}{2},b\neq 0,-1,-2,\dots$ ,
ⓘ 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, $s$ : nonnegative integer and $z$ : complex variable Referenced by: §13.6(iii), §13.6(iii), §8.7 Permalink: http://dlmf.nist.gov/13.11.E1 Encodings: TeX, pMML, png See also: Annotations for §13.11 and Ch.13

13.11.2		$M\left(a,b,z\right)=\Gamma\left(b-a-\tfrac{1}{2}\right){\mathrm{e}}^{\frac{1}{% 2}z}\left(\tfrac{1}{4}z\right)^{a-b+\frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\left(2b-2a-1\right)_{s}}{\left(b-2a\right)_{s}}(b-a-\frac{1}{2}+s)}{{% \left(b\right)_{s}}s!}I_{b-a-\frac{1}{2}+s}\left(\tfrac{1}{2}z\right),$
$b-a+\frac{1}{2},b\neq 0,-1,-2,\dots$ .
ⓘ 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, $s$ : nonnegative integer and $z$ : complex variable Proof sketch: Combine (13.24.1) with (13.14.4). A&S Ref: 13.3.6 Referenced by: §13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.11.E2 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.11 and Ch.13

(13.6.9), (13.6.11_1) and (13.6.11_2) are special cases.

13.11.3		${\mathbf{M}}\left(a,b,z\right)={\mathrm{e}}^{\frac{1}{2}z}\sum_{s=0}^{\infty}A% _{s}\left(b-2a\right)^{\frac{1}{2}(1-b-s)}\left(\tfrac{1}{2}z\right)^{\frac{1}% {2}(1-b+s)}J_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $s$ : nonnegative integer, $z$ : complex variable and $A_{n}$ : coefficients Source: Slater (1960, §3.8) A&S Ref: 13.3.7 Referenced by: §13.11, Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.11.E3 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.11 and Ch.13

where

13.11.4		$\displaystyle A_{0}$	$\displaystyle=1,$
		$\displaystyle A_{1}$	$\displaystyle=0,$
		$\displaystyle A_{2}$	$\displaystyle=\frac{1}{2}b,$
		$\displaystyle(n+1)A_{n+1}$	$\displaystyle=(n+b-1)A_{n-1}+(2a-b)A_{n-2},$
	$n=2,3,4,\dots$ .
ⓘ Defines: $A_{n}$ : coefficients (locally) Symbols: $n$ : nonnegative integer Referenced by: §13.11, Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.11.E4 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.11 and Ch.13

For additional expansions combine (13.14.4), (13.14.5), and §13.24. For other series expansions see Tricomi (1954, §1.8), Hansen (1975, §§66 and 87), Prudnikov et al. (1990, §6.6), López and Temme (2010a) and López and Pérez Sinusía (2014). See also §13.13.

13.10 Integrals 13.12 Products

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§13.11 Series

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