About the Project
13 Confluent Hypergeometric FunctionsKummer Functions

§13.13 Addition and Multiplication Theorems

Contents
  1. §13.13(i) Addition Theorems for M(a,b,z)
  2. §13.13(ii) Addition Theorems for U(a,b,z)
  3. §13.13(iii) Multiplication Theorems for M(a,b,z) and U(a,b,z)

§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)1bn=0(1b)n(y/x)nn!M(a,bn,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 eyn=0(ba)n(y)n(b)nn!M(a,b+n,x),
13.13.5 ey(xx+y)ban=0(ba)nynn!(x+y)nM(an,b,x),
((y+x)/x)>12,
13.13.6 ey(x+yx)1bn=0(1b)n(y)nn!xnM(an,bn,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)1bn=0(1+ab)n(y/x)nn!U(a,bn,x),
|y|<|x|,
13.13.9 (xx+y)an=0(a)n(1+ab)nynn!(x+y)nU(a+n,b,x),
(y/x)>12,
13.13.10 eyn=0(y)nn!U(a,b+n,x),
|y|<|x|,
13.13.11 ey(xx+y)ban=0(y)nn!(x+y)nU(an,b,x),
(y/x)>12,
13.13.12 ey(x+yx)1bn=0(y)nn!xnU(an,bn,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 (y1)x.