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35 Functions of Matrix ArgumentProperties

§35.6 Confluent Hypergeometric Functions of Matrix Argument

Contents

§35.6(i) Definitions

35.6.1 F11(ab;T)=k=01k!|κ|=k[a]κ[b]κZκ(T).
35.6.2 Ψ(a;b;T)=1Γm(a)Ωetr(-TX)|X|a-12(m+1)|I+X|b-a-12(m+1)dX,
(a)>12(m-1), TΩ.

Laguerre Form

35.6.3 Lν(γ)(T)=Γm(γ+ν+12(m+1))Γm(γ+12(m+1))F11(-νγ+12(m+1);T),
(γ),(γ+ν)>-1.

§35.6(ii) Properties

35.6.4 F11(ab;T)=1Bm(a,b-a)0<X<Ietr(TX)|X|a-12(m+1)|I-X|b-a-12(m+1)dX,
(a),(b-a)>12(m-1).
35.6.5 Ωetr(-TX)|X|b-12(m+1)F11(ab;SX)dX=Γm(b)|I-ST-1|-a|T|-b,
T>S, (b)>12(m-1).
35.6.6 Bm(b1,b2)|T|b1+b2-12(m+1)F11(a1+a2b1+b2;T)=0<X<T|X|b1-12(m+1)F11(a1b1;X)|T-X|b2-12(m+1)F11(a2b2;T-X)dX,
(b1),(b2)>12(m-1).
35.6.7 F11(ab;T)=etr(T)F11(b-ab;-T).
35.6.8 Ω|T|c-12(m+1)Ψ(a;b;T)dT=Γm(c)Γm(a-c)Γm(c-b+12(m+1))Γm(a)Γm(a-b+12(m+1)),
(a)>(c)+12(m-1)>m-1, (c-b)>-1.

§35.6(iii) Relations to Bessel Functions of Matrix Argument

§35.6(iv) Asymptotic Approximations

For asymptotic approximations for confluent hypergeometric functions of matrix argument, see Herz (1955) and Butler and Wood (2002).