13 Confluent Hypergeometric Functions Whittaker Functions 13.25 Products 13.27 Mathematical Applications

§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 $M_{\kappa,\mu}\left(z\right)$
§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 $W_{\kappa,\mu}\left(z\right)$

§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 $e^{-\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(-2\mu\right)_{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}% \*M_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(x\right),$

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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 ), : nonnegative integer, : real variable and : 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 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{{\left(1+2\mu\right)_{% n}}n!}\left(\frac{y}{\sqrt{x}}\right)^{n}\*M_{\kappa-\frac{1}{2}n,\mu+\frac{1}% {2}n}\left(x\right),$

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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 ), : nonnegative integer, : real variable and : 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 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu-\kappa\right)_{n}}y^{n}}{n!(x+y)^{n}}M_{\kappa-n,\mu}% \left(x\right),$ $\Re\left(y/x\right)>-\frac{1}{2}$ ,

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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 ), $\Re$ : real part, : nonnegative integer, : real variable and : 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 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{{\left(-2\mu\right)_{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*M_{% \kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\left(x\right),$

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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 ), : nonnegative integer, : real variable and : 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 $e^{\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{{\left(\frac{1}{2}+\mu+\kappa\right)_{n}}}{{\left(1+2\mu\right)_{n}}n!}% \left(\frac{-y}{\sqrt{x}}\right)^{n}\*M_{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}n}% \left(x\right),$

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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 ), : nonnegative integer, : real variable and : 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 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu+\kappa\right)_{n}}y^{n}}{n!(x+y)^{n}}M_{\kappa+n,\mu}% \left(x\right),$ $\Re\left((y+x)/x\right)>\frac{1}{2}$ .

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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 ), $\Re$ : real part, : nonnegative integer, : real variable and : 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 $e^{-\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}-\mu-\kappa\right)_{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*W_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(x\right),$

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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 ), : nonnegative integer, : real variable and : 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 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{% \infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*W_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(x\right),$

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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 ), : nonnegative integer, : real variable and : 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 $e^{-\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{{% \left(\frac{1}{2}+\mu-\kappa\right)_{n}}{\left(\frac{1}{2}-\mu-\kappa\right)_{% n}}}{n!}\*\left(\frac{y}{x+y}\right)^{n}W_{\kappa-n,\mu}\left(x\right),$ $\Re\left(y/x\right)>-\frac{1}{2}$ ,

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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 ), $\Re$ : real part, : nonnegative integer, : real variable and : 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 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\mu-\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*W_{\kappa+\frac{1}{2}n,\mu-% \frac{1}{2}n}\left(x\right),$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), : nonnegative integer, : real variable and : 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 $e^{\frac{1}{2}y}\left(\frac{x+y}{x}\right)^{\mu+\frac{1}{2}}\sum_{n=0}^{\infty% }\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*W_{\kappa+\frac{1}{2}n,\mu+% \frac{1}{2}n}\left(x\right),$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), : nonnegative integer, : real variable and : 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 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{1}% {n!}\left(\frac{-y}{x+y}\right)^{n}W_{\kappa+n,\mu}\left(x\right),$ $\Re\left(y/x\right)>-\frac{1}{2}$ .

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