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13 Confluent Hypergeometric FunctionsKummer Functions

§13.13 Addition and Multiplication Theorems

  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),
13.13.3 (xx+y)an=0(a)nynn!(x+y)nM(a+n,b,x),
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),
13.13.6 ey(x+yx)1bn=0(1b)n(y)nn!xnM(an,bn,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),
13.13.8 (x+yx)1bn=0(1+ab)n(y/x)nn!U(a,bn,x),
13.13.9 (xx+y)an=0(a)n(1+ab)nynn!(x+y)nU(a+n,b,x),
13.13.10 eyn=0(y)nn!U(a,b+n,x),
13.13.11 ey(xx+y)ban=0(y)nn!(x+y)nU(an,b,x),
13.13.12 ey(x+yx)1bn=0(y)nn!xnU(an,bn,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.