13 Confluent Hypergeometric Functions Kummer Functions 13.12 Products 13.14 Definitions and Basic Properties

§13.13 Addition and Multiplication Theorems

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Referenced by:: §13.11
Permalink:: http://dlmf.nist.gov/13.13
See also:: Annotations for Ch.13

§13.13(i) Addition Theorems for $M\left(a,b,z\right)$

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Keywords:: Kummer functions, addition theorems, series expansions
Notes:: See Erdélyi et al. (1953a, §6.14) and Slater (1960, §2.3). In the first reference Equation (2) needs the constraint $|\lambda-1|<1$ and Equation (6) should have no constraint. In the second reference (2.3.6) contains an error: $(x+y)^{n}$ should be replaced by $(x+y)^{-n}$ .
Permalink:: http://dlmf.nist.gov/13.13.i
See also:: Annotations for §13.13 and Ch.13

The function $M\left(a,b,x+y\right)$ has the following expansions:

13.13.1		$\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}y^{n}}{{\left(b\right)_{n}}n!}M% \left(a+n,b+n,x\right),$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E1 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.2		$\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{{\left(1-b\right)_{n}% }(-\ifrac{y}{x})^{n}}{n!}M\left(a,b-n,x\right),$
$\|y\|<\|x\|$ ,
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E2 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.3		$\left(\frac{x}{x+y}\right)^{a}\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}y^{% n}}{n!(x+y)^{n}}M\left(a+n,b,x\right),$
$\Re\left(y/x\right)>-\tfrac{1}{2}$ ,
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E3 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.4		$e^{y}\sum_{n=0}^{\infty}\frac{{\left(b-a\right)_{n}}(-y)^{n}}{{\left(b\right)_% {n}}n!}M\left(a,b+n,x\right),$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E4 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.5		$e^{y}\left(\frac{x}{x+y}\right)^{b-a}\sum_{n=0}^{\infty}\frac{{\left(b-a\right% )_{n}}y^{n}}{n!(x+y)^{n}}\*M\left(a-n,b,x\right),$
$\Re\left((y+x)/x\right)>\frac{1}{2}$ ,
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E5 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

13.13.6		$e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{{\left(1-b\right% )_{n}}(-y)^{n}}{n!x^{n}}\*M\left(a-n,b-n,x\right),$
$\|y\|<\|x\|$ .
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E6 Encodings: TeX, pMML, png See also: Annotations for §13.13(i), §13.13 and Ch.13

§13.13(ii) Addition Theorems for $U\left(a,b,z\right)$

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Keywords:: Kummer functions, addition theorems, series expansions
Notes:: See Slater (1960, §2.3.1).
Permalink:: http://dlmf.nist.gov/13.13.ii
See also:: Annotations for §13.13 and Ch.13

The function $U\left(a,b,x+y\right)$ has the following expansions:

13.13.7		$\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}(-y)^{n}}{n!}U\left(a+n,b+n,x% \right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E7 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.8		$\left(\frac{x+y}{x}\right)^{1-b}\*\sum_{n=0}^{\infty}\frac{{\left(1+a-b\right)% _{n}}(-\ifrac{y}{x})^{n}}{n!}U\left(a,b-n,x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E8 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.9		$\left(\frac{x}{x+y}\right)^{a}\sum_{n=0}^{\infty}\frac{{\left(a\right)_{n}}{% \left(1+a-b\right)_{n}}y^{n}}{n!(x+y)^{n}}U\left(a+n,b,x\right),$
$\Re\left(y/x\right)>-\tfrac{1}{2}$ ,
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E9 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.10		$e^{y}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!}U\left(a,b+n,x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E10 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.11		$e^{y}\left(\frac{x}{x+y}\right)^{b-a}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!(x+y% )^{n}}U\left(a-n,b,x\right),$
$\Re\left(y/x\right)>-\tfrac{1}{2}$ ,
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E11 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

13.13.12		$e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!x^{n% }}U\left(a-n,b-n,x\right),$
$\|y\|<\|x\|$ .
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/13.13.E12 Encodings: TeX, pMML, png See also: Annotations for §13.13(ii), §13.13 and Ch.13

§13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$

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Keywords:: Kummer functions, multiplication theorems, series expansions
Notes:: See Slater (1960, §§2.3.2, 2.3.3).
Permalink:: http://dlmf.nist.gov/13.13.iii
See also:: Annotations for §13.13 and Ch.13

To obtain similar expansions for $M\left(a,b,xy\right)$ and $U\left(a,b,xy\right)$ , replace $y$ in the previous two subsections by $(y-1)x$ .

13.12 Products 13.14 Definitions and Basic Properties

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§13.13 Addition and Multiplication Theorems

Contents

§13.13(i) Addition Theorems for $M\left(a,b,z\right)$

§13.13(ii) Addition Theorems for $U\left(a,b,z\right)$

§13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$

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§13.13 Addition and Multiplication Theorems

Contents

§13.13(i) Addition Theorems for M⁡(a,b,z)

§13.13(ii) Addition Theorems for U⁡(a,b,z)

§13.13(iii) Multiplication Theorems for M⁡(a,b,z) and U⁡(a,b,z)

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§13.13(i) Addition Theorems for $M\left(a,b,z\right)$

§13.13(ii) Addition Theorems for $U\left(a,b,z\right)$

§13.13(iii) Multiplication Theorems for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$