§13.13 Addition and Multiplication Theorems

Referenced by:: §13.11
Permalink:: http://dlmf.nist.gov/13.13

§13.13(i) Addition Theorems for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$
§13.13(ii) Addition Theorems for $\mathop{U\/}\nolimits\!\left(a,b,z\right)$
§13.13(iii) Multiplication Theorems for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,z\right)$

§13.13(i) Addition Theorems for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$

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 $(x+y)^{{-n}}$ .
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.13.i

The function $\mathop{M\/}\nolimits\!\left(a,b,x+y\right)$ has the following expansions:

13.13.1 $\sum _{{n=0}}^{{\infty}}\frac{\left(a\right)_{{n}}y^{n}}{\left(b\right)_{{n}}n!}\mathop{M\/}\nolimits\!\left(a+n,b+n,x\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E1
Encodings:: TeX, pMML, png

13.13.2 $\left(\frac{x+y}{x}\right)^{{1-b}}\sum _{{n=0}}^{{\infty}}\frac{\left(1-b\right)_{{n}}(-\ifrac{y}{x})^{n}}{n!}\mathop{M\/}\nolimits\!\left(a,b-n,x\right),$ $|y|<|x|$ ,

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E2
Encodings:: TeX, pMML, png

13.13.3 $\left(\frac{x}{x+y}\right)^{{a}}\sum _{{n=0}}^{{\infty}}\frac{\left(a\right)_{{n}}y^{n}}{n!(x+y)^{{n}}}\mathop{M\/}\nolimits\!\left(a+n,b,x\right),$ $\realpart{(y/x)}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\realpart{}$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E3
Encodings:: TeX, pMML, png

13.13.4 $e^{{y}}\sum _{{n=0}}^{{\infty}}\frac{\left(b-a\right)_{{n}}(-y)^{n}}{\left(b\right)_{{n}}n!}\mathop{M\/}\nolimits\!\left(a,b+n,x\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E4
Encodings:: TeX, pMML, png

13.13.5 $e^{{y}}\left(\frac{x}{x+y}\right)^{{b-a}}\sum _{{n=0}}^{{\infty}}\frac{\left(b-a\right)_{{n}}y^{n}}{n!(x+y)^{n}}\*\mathop{M\/}\nolimits\!\left(a-n,b,x\right),$ $\realpart{((y+x)/x)}>\frac{1}{2}$ ,

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\realpart{}$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E5
Encodings:: TeX, pMML, png

13.13.6 $e^{{y}}\left(\frac{x+y}{x}\right)^{{1-b}}\sum _{{n=0}}^{{\infty}}\frac{\left(1-b\right)_{{n}}(-y)^{n}}{n!x^{{n}}}\*\mathop{M\/}\nolimits\!\left(a-n,b-n,x\right),$ $|y|<|x|$ .

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E6
Encodings:: TeX, pMML, png

§13.13(ii) Addition Theorems for $\mathop{U\/}\nolimits\!\left(a,b,z\right)$

Notes:: See Slater (1960, §2.3.1).
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.13.ii

The function $\mathop{U\/}\nolimits\!\left(a,b,x+y\right)$ has the following expansions:

13.13.7 $\sum _{{n=0}}^{{\infty}}\frac{\left(a\right)_{{n}}(-y)^{n}}{n!}\mathop{U\/}\nolimits\!\left(a+n,b+n,x\right),$ $|y|<|x|$ ,

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E7
Encodings:: TeX, pMML, png

13.13.8 $\left(\frac{x+y}{x}\right)^{{1-b}}\*\sum _{{n=0}}^{{\infty}}\frac{\left(1+a-b\right)_{{n}}(-\ifrac{y}{x})^{n}}{n!}\mathop{U\/}\nolimits\!\left(a,b-n,x\right),$ $|y|<|x|$ ,

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E8
Encodings:: TeX, pMML, png

13.13.9 $\left(\frac{x}{x+y}\right)^{{a}}\sum _{{n=0}}^{{\infty}}\frac{\left(a\right)_{{n}}\left(1+a-b\right)_{{n}}y^{n}}{n!(x+y)^{{n}}}\mathop{U\/}\nolimits\!\left(a+n,b,x\right),$ $\realpart{(y/x)}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\realpart{}$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E9
Encodings:: TeX, pMML, png

13.13.10 $e^{{y}}\sum _{{n=0}}^{{\infty}}\frac{(-y)^{n}}{n!}\mathop{U\/}\nolimits\!\left(a,b+n,x\right),$ $|y|<|x|$ ,

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E10
Encodings:: TeX, pMML, png

13.13.11 $e^{{y}}\left(\frac{x}{x+y}\right)^{{b-a}}\sum _{{n=0}}^{{\infty}}\frac{(-y)^{n}}{n!(x+y)^{n}}\mathop{U\/}\nolimits\!\left(a-n,b,x\right),$ $\realpart{(y/x)}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\realpart{}$ : real part, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E11
Encodings:: TeX, pMML, png

13.13.12 $e^{{y}}\left(\frac{x+y}{x}\right)^{{1-b}}\sum _{{n=0}}^{{\infty}}\frac{(-y)^{n}}{n!x^{{n}}}\mathop{U\/}\nolimits\!\left(a-n,b-n,x\right),$ $|y|<|x|$ .

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/13.13.E12
Encodings:: TeX, pMML, png

§13.13(iii) Multiplication Theorems for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,z\right)$

Notes:: See Slater (1960, §§2.3.2, 2.3.3).
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.13.iii

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

§13.13 Addition and Multiplication Theorems

Contents

§13.13(i) Addition Theorems for

§13.13(ii) Addition Theorems for

§13.13(iii) Multiplication Theorems for and

§13.13(i) Addition Theorems for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$

§13.13(ii) Addition Theorems for $\mathop{U\/}\nolimits\!\left(a,b,z\right)$

§13.13(iii) Multiplication Theorems for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,z\right)$