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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.26 Addition and Multiplication Theorems

Contents
  1. §13.26(i) Addition Theorems for Mκ,μ(z)
  2. §13.26(ii) Addition Theorems for Wκ,μ(z)
  3. §13.26(iii) Multiplication Theorems for Mκ,μ(z) and Wκ,μ(z)

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

The function Mκ,μ(x+y) has the following expansions:

13.26.1 e12y(xx+y)μ12n=0(2μ)nn!(yx)nMκ12n,μ12n(x),
|y|<|x|,
13.26.2 e12y(x+yx)μ+12n=0(12+μκ)n(1+2μ)nn!(yx)nMκ12n,μ+12n(x),
13.26.3 e12y(x+yx)κn=0(12+μκ)nynn!(x+y)nMκn,μ(x),
(y/x)>12,
13.26.4 e12y(xx+y)μ12n=0(2μ)nn!(yx)nMκ+12n,μ12n(x),
|y|<|x|,
13.26.5 e12y(x+yx)μ+12n=0(12+μ+κ)n(1+2μ)nn!(yx)nMκ+12n,μ+12n(x),
13.26.6 e12y(xx+y)κn=0(12+μ+κ)nynn!(x+y)nMκ+n,μ(x),
((y+x)/x)>12.

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

The function Wκ,μ(x+y) has the following expansions:

13.26.7 e12y(xx+y)μ12n=0(12μκ)nn!(yx)nWκ12n,μ12n(x),
|y|<|x|,
13.26.8 e12y(x+yx)μ+12n=0(12+μκ)nn!(yx)nWκ12n,μ+12n(x),
|y|<|x|,
13.26.9 e12y(x+yx)κn=0(12+μκ)n(12μκ)nn!(yx+y)nWκn,μ(x),
(y/x)>12,
13.26.10 e12y(xx+y)μ12n=01n!(yx)nWκ+12n,μ12n(x),
|y|<|x|,
13.26.11 e12y(x+yx)μ+12n=01n!(yx)nWκ+12n,μ+12n(x),
|y|<|x|,
13.26.12 e12y(xx+y)κn=01n!(yx+y)nWκ+n,μ(x),
(y/x)>12.

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

To obtain similar expansions for Mκ,μ(xy) and Wκ,μ(xy), replace y in the previous two subsections by (y1)x.