13 Confluent Hypergeometric Functions Whittaker Functions 13.25 Products 13.27 Mathematical Applications

§13.26 Addition and Multiplication Theorems

Referenced by:: §13.24(ii)
Permalink:: http://dlmf.nist.gov/13.26

§13.26(i) Addition Theorems for $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$
§13.26(ii) Addition Theorems for $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$
§13.26(iii) Multiplication Theorems for $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ and $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$

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

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.
Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.26.i

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

13.26.1 $e^{{-\frac{1}{2}y}}\left(\frac{x}{x+y}\right)^{{\mu-\frac{1}{2}}}\sum_{{n=0}}^% {{\infty}}\frac{\left(-2\mu\right)_{{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^% {n}\*\mathop{M_{{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(x% \right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E1
Encodings:: TeX, pMML, png

13.26.2 $e^{{-\frac{1}{2}y}}\left(\frac{x+y}{x}\right)^{{\mu+\frac{1}{2}}}\sum_{{n=0}}^% {{\infty}}\frac{\left(\frac{1}{2}+\mu-\kappa\right)_{{n}}}{\left(1+2\mu\right)% _{{n}}n!}\left(\frac{y}{\sqrt{x}}\right)^{n}\*\mathop{M_{{\kappa-\frac{1}{2}n,% \mu+\frac{1}{2}n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E2
Encodings:: TeX, pMML, png

13.26.3 $e^{{-\frac{1}{2}y}}\left(\frac{x+y}{x}\right)^{{\kappa}}\sum_{{n=0}}^{{\infty}% }\frac{\left(\frac{1}{2}+\mu-\kappa\right)_{{n}}y^{n}}{n!(x+y)^{n}}\mathop{M_{% {\kappa-n,\mu}}\/}\nolimits\!\left(x\right),$ $\realpart{(y/x)}>-\frac{1}{2}$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\realpart{}$ : 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

13.26.4 $e^{{\frac{1}{2}y}}\left(\frac{x}{x+y}\right)^{{\mu-\frac{1}{2}}}\sum_{{n=0}}^{% {\infty}}\frac{\left(-2\mu\right)_{{n}}}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{% n}\*\mathop{M_{{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(x% \right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E4
Encodings:: TeX, pMML, png

13.26.5 $e^{{\frac{1}{2}y}}\left(\frac{x+y}{x}\right)^{{\mu+\frac{1}{2}}}\sum_{{n=0}}^{% {\infty}}\frac{\left(\frac{1}{2}+\mu+\kappa\right)_{{n}}}{\left(1+2\mu\right)_% {{n}}n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*\mathop{M_{{\kappa+\frac{1}{2}n,% \mu+\frac{1}{2}n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E5
Encodings:: TeX, pMML, png

13.26.6 $e^{{\frac{1}{2}y}}\left(\frac{x}{x+y}\right)^{{\kappa}}\sum_{{n=0}}^{{\infty}}% \frac{\left(\frac{1}{2}+\mu+\kappa\right)_{{n}}y^{n}}{n!(x+y)^{n}}\mathop{M_{{% \kappa+n,\mu}}\/}\nolimits\!\left(x\right),$ $\realpart{((y+x)/x)}>\frac{1}{2}$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\realpart{}$ : 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

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

Notes:: See Slater (1960, §2.6.1).
Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.26.ii

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

13.26.7 $e^{{-\frac{1}{2}y}}\left(\frac{x}{x+y}\right)^{{\mu-\frac{1}{2}}}\sum_{{n=0}}^% {{\infty}}\frac{\left(\frac{1}{2}-\mu-\kappa\right)_{{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*\mathop{W_{{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}% \nolimits\!\left(x\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E7
Encodings:: TeX, pMML, png

13.26.8 $e^{{-\frac{1}{2}y}}\left(\frac{x+y}{x}\right)^{{\mu+\frac{1}{2}}}\sum_{{n=0}}^% {{\infty}}\frac{\left(\frac{1}{2}+\mu-\kappa\right)_{{n}}}{n!}\left(\frac{-y}{% \sqrt{x}}\right)^{n}\*\mathop{W_{{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}}\/}% \nolimits\!\left(x\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E8
Encodings:: TeX, pMML, png

13.26.9 $e^{{-\frac{1}{2}y}}\left(\frac{x+y}{x}\right)^{{\kappa}}\sum_{{n=0}}^{{\infty}% }\frac{\left(\frac{1}{2}+\mu-\kappa\right)_{{n}}\left(\frac{1}{2}-\mu-\kappa% \right)_{{n}}}{n!}\*\left(\frac{y}{x+y}\right)^{n}\mathop{W_{{\kappa-n,\mu}}\/% }\nolimits\!\left(x\right),$ $\realpart{(y/x)}>-\frac{1}{2}$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\realpart{}$ : real part, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E9
Encodings:: TeX, pMML, png

13.26.10 $e^{{\frac{1}{2}y}}\left(\frac{x}{x+y}\right)^{{\mu-\frac{1}{2}}}\sum_{{n=0}}^{% {\infty}}\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*\mathop{W_{{\kappa+% \frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E10
Encodings:: TeX, pMML, png

13.26.11 $e^{{\frac{1}{2}y}}\left(\frac{x+y}{x}\right)^{{\mu+\frac{1}{2}}}\sum_{{n=0}}^{% {\infty}}\frac{1}{n!}\left(\frac{-y}{\sqrt{x}}\right)^{n}\*\mathop{W_{{\kappa+% \frac{1}{2}n,\mu+\frac{1}{2}n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E11
Encodings:: TeX, pMML, png

13.26.12 $e^{{\frac{1}{2}y}}\left(\frac{x}{x+y}\right)^{{\kappa}}\sum_{{n=0}}^{{\infty}}% \frac{1}{n!}\left(\frac{-y}{x+y}\right)^{n}\mathop{W_{{\kappa+n,\mu}}\/}% \nolimits\!\left(x\right),$ $\realpart{(y/x)}>-\frac{1}{2}$ .

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\realpart{}$ : real part, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/13.26.E12
Encodings:: TeX, pMML, png

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

Notes:: See Slater (1960, §§2.6.2, 2.6.3).
Keywords:: Whittaker functions
Permalink:: http://dlmf.nist.gov/13.26.iii

To obtain similar expansions for $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(xy\right)$ and $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(xy\right)$ , replace in the previous two subsections by .

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 13.25 Products 13.27 Mathematical Applications

§13.26 Addition and Multiplication Theorems

Contents

§13.26(i) Addition Theorems for

§13.26(ii) Addition Theorems for

§13.26(iii) Multiplication Theorems for and

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

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

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