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13 Confluent Hypergeometric FunctionsKummer Functions

§13.6 Relations to Other Functions

  1. §13.6(i) Elementary Functions
  2. §13.6(ii) Incomplete Gamma Functions
  3. §13.6(iii) Modified Bessel Functions
  4. §13.6(iv) Parabolic Cylinder Functions
  5. §13.6(v) Orthogonal Polynomials
  6. §13.6(vi) Generalized Hypergeometric Functions
  7. §13.6(vii) Coulomb Functions

§13.6(i) Elementary Functions

13.6.1 M(a,a,z) =ez,
13.6.2 M(1,2,2z) =ezzsinhz,
13.6.3 M(0,b,z) =U(0,b,z)=1,
13.6.4 U(a,a+1,z) =za.

§13.6(ii) Incomplete Gamma Functions

For the notation see §§6.2(i), 7.2(i), 8.2(i), and 8.19(i). When ab is an integer or a is a positive integer the Kummer functions can be expressed as incomplete gamma functions (or generalized exponential integrals). For example,

13.6.5 M(a,a+1,z)=ezM(1,a+1,z)=azaγ(a,z),
13.6.6 U(a,a,z)=z1aU(1,2a,z)=z1aezEa(z)=ezΓ(1a,z).

Special cases are the error functions

13.6.7 M(12,32,z2)=π2zerf(z),
13.6.8 U(12,12,z2)=πez2erfc(z).

§13.6(iii) Modified Bessel Functions

When b=2a the Kummer functions can be expressed as modified Bessel functions. For the notation see §§10.25(ii) and 9.2(i).

13.6.9 M(ν+12,2ν+1,2z) =Γ(1+ν)ez(z/2)νIν(z),
13.6.10 U(ν+12,2ν+1,2z) =1πez(2z)νKν(z),
13.6.11 U(56,53,43z3/2) =π35/6exp(23z3/2)22/3zAi(z),

and in the case that b2a is an integer we have

13.6.11_1 M(ν+12,2ν+1+n,2z) =Γ(ν)ez(z/2)νk=0n(n)k(2ν)k(ν+k)(2ν+1+n)kk!Iν+k(z),
13.6.11_2 M(ν+12,2ν+1n,2z) =Γ(νn)ez(z/2)nνk=0n(1)k(n)k(2ν2n)k(νn+k)(2ν+1n)kk!Iν+kn(z).

Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively

§13.6(iv) Parabolic Cylinder Functions

For the notation see §12.2.

13.6.12 U(12a+14,12,12z2)=212a+14e14z2U(a,z),
13.6.13 U(12a+34,32,12z2)=212a+34e14z2zU(a,z).
13.6.14 M(12a+14,12,12z2)=212a34Γ(12a+34)e14z2π(U(a,z)+U(a,z)),
13.6.15 M(12a+34,32,12z2)=212a54Γ(12a+14)e14z2zπ(U(a,z)U(a,z)).

§13.6(v) Orthogonal Polynomials

Special cases of §13.6(iv) are as follows. For the notation see §§18.3, 18.19.

Hermite Polynomials

13.6.16 M(n,12,z2) =(1)nn!(2n)!H2n(z),
13.6.17 M(n,32,z2) =(1)nn!(2n+1)!2zH2n+1(z),
13.6.18 U(1212n,32,z2) =2nz1Hn(z).

Laguerre Polynomials

Charlier Polynomials

§13.6(vi) Generalized Hypergeometric Functions

13.6.21 U(a,b,z)=zaF02(a,ab+1;;z1).

For the definition of F02(a,ab+1;;z1) when neither a nor ab+1 is a nonpositive integer see §16.5.

§13.6(vii) Coulomb Functions

For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).