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13 Confluent Hypergeometric FunctionsKummer Functions

§13.6 Relations to Other 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) =z-a.

§13.6(ii) Incomplete Gamma Functions

For the notation see §§6.2(i), 7.2(i), 8.2(i), and 8.19(i). When a-b 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)=e-zM(1,a+1,z)=az-aγ(a,z),
13.6.6 U(a,a,z)=z1-aU(1,2-a,z)=z1-aezEa(z)=ezΓ(1-a,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 b-2a 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+1-n,2z) =Γ(ν-n)ez(z/2)n-νk=0n(-1)k(-n)k(2ν-2n)k(ν-n+k)(2ν+1-n)kk!Iν+k-n(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)=212a-34Γ(12a+34)e14z2π(U(a,z)+U(a,-z)),
13.6.15 M(12a+34,32,12z2)=212a-54Γ(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(12-12n,32,z2) =2-nz-1Hn(z).

Laguerre Polynomials

Charlier Polynomials

§13.6(vi) Generalized Hypergeometric Functions

13.6.21 U(a,b,z)=z-aF02(a,a-b+1;-;-z-1).

For the definition of F02(a,a-b+1;-;-z-1) when neither a nor a-b+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).