§13.26 Addition and Multiplication Theorems

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Referenced by:: §13.24(ii)
Permalink:: http://dlmf.nist.gov/13.26
See also:: Annotations for Ch.13

§13.26(i) Addition Theorems for $M_{\kappa,\mu}\left(z\right)$

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Keywords:: Whittaker functions, addition theorems
Notes:: See Slater (1960, §2.6). Note that (2.6.3) and (2.6.6) contain errors; the correct versions are (13.26.3) and (13.26.6), respectively.
Permalink:: http://dlmf.nist.gov/13.26.i
See also:: Annotations for §13.26 and Ch.13

The function $M_{\kappa,\mu}\left(x+y\right)$ has the following expansions:

13.26.1		$e^{-\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(-2\mu\right)_{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}% \*M_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E1 Encodings: TeX, pMML, png See also: Annotations for §13.26(i), §13.26 and Ch.13
13.26.2		$e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{{\left(1+2\mu\right)_{% n}}n!}\left(\frac{y}{\sqrt{x}}\right)^{n}\*M_{\kappa-\frac{1}{2}n,\mu+\frac{1}% {2}n}\left(x\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E2 Encodings: TeX, pMML, png See also: Annotations for §13.26(i), §13.26 and Ch.13
13.26.3		$e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu-\kappa\right)_{n}}y^{n}}{n!(x+y)^{n}}M_{\kappa-n,\mu}% \left(x\right),$
$\Re\left(y/x\right)>-\frac{1}{2}$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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 Referenced by: §13.26(i) Permalink: http://dlmf.nist.gov/13.26.E3 Encodings: TeX, pMML, png See also: Annotations for §13.26(i), §13.26 and Ch.13
13.26.4		$e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{{\left(-2\mu\right)_{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*M_{% \kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\left(x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E4 Encodings: TeX, pMML, png See also: Annotations for §13.26(i), §13.26 and Ch.13
13.26.5		$e^{\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{{\left(\frac{1}{2}+\mu+\kappa\right)_{n}}}{{\left(1+2\mu\right)_{n}}n!}% \left(\frac{-y}{\sqrt{x}}\right)^{n}\*M_{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}n}% \left(x\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E5 Encodings: TeX, pMML, png See also: Annotations for §13.26(i), §13.26 and Ch.13
13.26.6		$e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu+\kappa\right)_{n}}y^{n}}{n!(x+y)^{n}}M_{\kappa+n,\mu}% \left(x\right),$
$\Re\left((y+x)/x\right)>\frac{1}{2}$ .
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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 Referenced by: §13.26(i) Permalink: http://dlmf.nist.gov/13.26.E6 Encodings: TeX, pMML, png See also: Annotations for §13.26(i), §13.26 and Ch.13

§13.26(ii) Addition Theorems for $W_{\kappa,\mu}\left(z\right)$

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Keywords:: Whittaker functions, addition theorems, series expansions
Notes:: See Slater (1960, §2.6.1).
Permalink:: http://dlmf.nist.gov/13.26.ii
See also:: Annotations for §13.26 and Ch.13

The function $W_{\kappa,\mu}\left(x+y\right)$ has the following expansions:

13.26.7		$e^{-\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}-\mu-\kappa\right)_{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*W_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E7 Encodings: TeX, pMML, png See also: Annotations for §13.26(ii), §13.26 and Ch.13
13.26.8		$e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*W_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E8 Encodings: TeX, pMML, png See also: Annotations for §13.26(ii), §13.26 and Ch.13
13.26.9		$e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu-\kappa\right)_{n}}{\left(\frac{1}{2}-\mu-\kappa\right)_{% n}}}{n!}\*\left(\frac{y}{x+y}\right)^{n}W_{\kappa-n,\mu}\left(x\right),$
$\Re\left(y/x\right)>-\frac{1}{2}$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E9 Encodings: TeX, pMML, png See also: Annotations for §13.26(ii), §13.26 and Ch.13
13.26.10		$e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*W_{\kappa+\frac{1}{2}n,\mu-% \frac{1}{2}n}\left(x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E10 Encodings: TeX, pMML, png See also: Annotations for §13.26(ii), §13.26 and Ch.13
13.26.11		$e^{\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*W_{\kappa+\frac{1}{2}n,\mu+% \frac{1}{2}n}\left(x\right),$
$\|y\|<\|x\|$ ,
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E11 Encodings: TeX, pMML, png See also: Annotations for §13.26(ii), §13.26 and Ch.13
13.26.12		$e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{1}% {n!}\left(\frac{-y}{x+y}\right)^{n}W_{\kappa+n,\mu}\left(x\right),$
$\Re\left(y/x\right)>-\frac{1}{2}$ .
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker 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/13.26.E12 Encodings: TeX, pMML, png See also: Annotations for §13.26(ii), §13.26 and Ch.13

§13.26(iii) Multiplication Theorems for $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$

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Keywords:: Whittaker functions, multiplication theorems, series expansions
Notes:: See Slater (1960, §§2.6.2, 2.6.3).
Permalink:: http://dlmf.nist.gov/13.26.iii
See also:: Annotations for §13.26 and Ch.13

To obtain similar expansions for $M_{\kappa,\mu}\left(xy\right)$ and $W_{\kappa,\mu}\left(xy\right)$ , replace $y$ in the previous two subsections by $(y-1)x$ .

§13.26 Addition and Multiplication Theorems

Contents

§13.26(i) Addition Theorems for $M_{\kappa,\mu}\left(z\right)$

§13.26(ii) Addition Theorems for $W_{\kappa,\mu}\left(z\right)$

§13.26(iii) Multiplication Theorems for $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$

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§13.26 Addition and Multiplication Theorems

Contents

§13.26(i) Addition Theorems for Mκ,μ⁡(z)

§13.26(ii) Addition Theorems for Wκ,μ⁡(z)

§13.26(iii) Multiplication Theorems for Mκ,μ⁡(z) and Wκ,μ⁡(z)

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§13.26(i) Addition Theorems for $M_{\kappa,\mu}\left(z\right)$

§13.26(ii) Addition Theorems for $W_{\kappa,\mu}\left(z\right)$

§13.26(iii) Multiplication Theorems for $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$