Digital Library of Mathematical Functions
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13 Confluent Hypergeometric FunctionsKummer Functions

§13.13 Addition and Multiplication Theorems

Contents

§13.13(i) Addition Theorems for M(a,b,z)

The function M(a,b,x+y) has the following expansions:

13.13.1 n=0(a)nyn(b)nn!M(a+n,b+n,x),
13.13.2 (x+yx)1-bn=0(1-b)n(-y/x)nn!M(a,b-n,x),
|y|<|x|,
13.13.3 (xx+y)an=0(a)nynn!(x+y)nM(a+n,b,x),
(y/x)>-12,
13.13.4 yn=0(b-a)n(-y)n(b)nn!M(a,b+n,x),
13.13.5 y(xx+y)b-an=0(b-a)nynn!(x+y)nM(a-n,b,x),
((y+x)/x)>12,
13.13.6 y(x+yx)1-bn=0(1-b)n(-y)nn!xnM(a-n,b-n,x),
|y|<|x|.

§13.13(ii) Addition Theorems for U(a,b,z)

The function U(a,b,x+y) has the following expansions:

13.13.7 n=0(a)n(-y)nn!U(a+n,b+n,x),
|y|<|x|,
13.13.8 (x+yx)1-bn=0(1+a-b)n(-y/x)nn!U(a,b-n,x),
|y|<|x|,
13.13.9 (xx+y)an=0(a)n(1+a-b)nynn!(x+y)nU(a+n,b,x),
(y/x)>-12,
13.13.10 yn=0(-y)nn!U(a,b+n,x),
|y|<|x|,
13.13.11 y(xx+y)b-an=0(-y)nn!(x+y)nU(a-n,b,x),
(y/x)>-12,
13.13.12 y(x+yx)1-bn=0(-y)nn!xnU(a-n,b-n,x),
|y|<|x|.

§13.13(iii) Multiplication Theorems for M(a,b,z) and U(a,b,z)

To obtain similar expansions for M(a,b,xy) and U(a,b,xy), replace y in the previous two subsections by (y-1)x.