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
§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(vii) Coulomb Functions

§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 $M\left(a,a,z\right)=e^{z},$

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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 : 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 $M\left(1,2,2z\right)=\frac{e^{z}}{z}\sinh z,$

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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 : 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 $M\left(0,b,z\right)=U\left(0,b,z\right)=1,$

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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 : 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 $U\left(a,a+1,z\right)=z^{-a}.$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 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 $M\left(a,a+1,-z\right)=e^{-z}M\left(1,a+1,z\right)=az^{-a}\gamma\left(a,z% \right),$

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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 : 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 $U\left(a,a,z\right)=z^{1-a}U\left(1,2-a,z\right)=z^{1-a}e^{z}E_{a}\left(z% \right)=e^{z}\Gamma\left(1-a,z\right).$

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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 : 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 $M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)=\frac{\sqrt{\pi}}{2z}% \operatorname{erf}\left(z\right),$

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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 : 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 $U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)=\sqrt{\pi}e^{z^{2}}\operatorname% {erfc}\left(z\right).$

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

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

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

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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 : 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 $U\left(\tfrac{5}{6},\tfrac{5}{3},\tfrac{4}{3}z^{3/2}\right)=\sqrt{\pi}\frac{3^% {5/6}\exp\left(\tfrac{2}{3}z^{3/2}\right)}{2^{2/3}z}\operatorname{Ai}\left(z% \right),$

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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 : 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 is an integer we have

13.6.11_1 $M\left(\nu+\tfrac{1}{2},2\nu+1+n,2z\right)=\Gamma\left(\nu\right){\mathrm{e}}^% {z}\left(\ifrac{z}{2}\right)^{-\nu}\sum_{k=0}^{n}\frac{{\left(-n\right)_{k}}{% \left(2\nu\right)_{k}}(\nu+k)}{{\left(2\nu+1+n\right)_{k}}k!}I_{\nu+k}\left(z% \right),$

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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 ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : nonnegative integer and : 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 $M\left(\nu+\tfrac{1}{2},2\nu+1-n,2z\right)=\Gamma\left(\nu-n\right){\mathrm{e}% }^{z}\left(\ifrac{z}{2}\right)^{n-\nu}\sum_{k=0}^{n}(-1)^{k}\frac{{\left(-n% \right)_{k}}{\left(2\nu-2n\right)_{k}}(\nu-n+k)}{{\left(2\nu+1-n\right)_{k}}k!% }I_{\nu+k-n}\left(z\right).$

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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 ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : nonnegative integer and : 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 $U\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=2^{% \frac{1}{2}a+\frac{1}{4}}e^{\frac{1}{4}z^{2}}U\left(a,z\right),$

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

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

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

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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 : 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 .) 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 (§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 $M\left(-n,\tfrac{1}{2},z^{2}\right)=(-1)^{n}\frac{n!}{(2n)!}H_{2n}\left(z% \right),$

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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 ), : 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
See also:: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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

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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 ), : 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
See also:: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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

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Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : 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
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 $U\left(-n,\alpha+1,z\right)=(-1)^{n}{\left(\alpha+1\right)_{n}}M\left(-n,% \alpha+1,z\right)=(-1)^{n}n!L^{(\alpha)}_{n}\left(z\right).$

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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 ), : nonnegative integer and : 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 $U\left(-n,z-n+1,a\right)={\left(-z\right)_{n}}M\left(-n,z-n+1,a\right)=a^{n}C_% {n}\left(z;a\right).$

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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), : nonnegative integer and : 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 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.
Permalink:: http://dlmf.nist.gov/13.6.vi
See also:: Annotations for §13.6 and Ch.13