Digital Library of Mathematical Functions
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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.26 Addition and Multiplication Theorems

Contents

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

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

13.26.1 -12y(xx+y)μ-12n=0(-2μ)nn!(-yx)nMκ-12n,μ-12n(x),
|y|<|x|,
13.26.2 -12y(x+yx)μ+12n=0(12+μ-κ)n(1+2μ)nn!(yx)nMκ-12n,μ+12n(x),
13.26.3 -12y(x+yx)κn=0(12+μ-κ)nynn!(x+y)nMκ-n,μ(x),
(y/x)>-12,
13.26.4 12y(xx+y)μ-12n=0(-2μ)nn!(-yx)nMκ+12n,μ-12n(x),
|y|<|x|,
13.26.5 12y(x+yx)μ+12n=0(12+μ+κ)n(1+2μ)nn!(-yx)nMκ+12n,μ+12n(x),
13.26.6 12y(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 -12y(xx+y)μ-12n=0(12-μ-κ)nn!(-yx)nWκ-12n,μ-12n(x),
|y|<|x|,
13.26.8 -12y(x+yx)μ+12n=0(12+μ-κ)nn!(-yx)nWκ-12n,μ+12n(x),
|y|<|x|,
13.26.9 -12y(x+yx)κn=0(12+μ-κ)n(12-μ-κ)nn!(yx+y)nWκ-n,μ(x),
(y/x)>-12,
13.26.10 12y(xx+y)μ-12n=01n!(-yx)nWκ+12n,μ-12n(x),
|y|<|x|,
13.26.11 12y(x+yx)μ+12n=01n!(-yx)nWκ+12n,μ+12n(x),
|y|<|x|,
13.26.12 12y(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 (y-1)x.