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35 Functions of Matrix ArgumentProperties

§35.8 Generalized Hypergeometric Functions of Matrix Argument

Contents

§35.8(i) Definition

Let p and q be nonnegative integers; a1,,ap; b1,,bq; -bj+12(k+1), 1jq, 1km. The generalized hypergeometric function Fqp with matrix argument T𝒮, numerator parameters a1,,ap, and denominator parameters b1,,bq is

35.8.1 Fqp(a1,,apb1,,bq;T)=k=01k!|κ|=k[a1]κ[ap]κ[b1]κ[bq]κZκ(T).

Convergence Properties

If -aj+12(k+1) for some j,k satisfying 1jp, 1km, then the series expansion (35.8.1) terminates.

If pq, then (35.8.1) converges for all T.

If p=q+1, then (35.8.1) converges absolutely for ||T||<1 and diverges for ||T||>1.

If p>q+1, then (35.8.1) diverges unless it terminates.

§35.8(ii) Relations to Other Functions

§35.8(iii) F23 Case

Kummer Transformation

Let c=b1+b2-a1-a2-a3. Then

35.8.5 F23(a1,a2,a3b1,b2;I)=Γm(b2)Γm(c)Γm(b2-a3)Γm(c+a3)F23(b1-a1,b1-a2,a3b1,c+a3;I),
(b2),(c)>12(m-1).

Pfaff–Saalschutz Formula

Let a1+a2+a3+12(m+1)=b1+b2; one of the aj be a negative integer; (b1-a1), (b1-a2), (b1-a3), (b1-a1-a2-a3)>12(m-1). Then

35.8.6 F23(a1,a2,a3b1,b2;I)=Γm(b1-a1)Γm(b1-a2)Γm(b1)Γm(b1-a1-a2)Γm(b1-a3)Γm(b1-a1-a2-a3)Γm(b1-a1-a3)Γm(b1-a2-a3).

Thomae Transformation

Again, let c=b1+b2-a1-a2-a3. Then

35.8.7 F23(a1,a2,a3b1,b2;I)=Γm(b1)Γm(b2)Γ(c)Γm(a1)Γm(c+a2)Γ(c+a3)F23(b1-a1,b2-a2,cc+a2,c+a3;I),
(b1), (b2), (c)>12(m-1).

§35.8(iv) General Properties

Value at T=0

Confluence

35.8.9 limγFqp+1(a1,,ap,γb1,,bq;γ-1T)=Fqp(a1,,apb1,,bq;T),
35.8.10 limγFq+1p(a1,,apb1,,bq,γ;γT)=Fqp(a1,,apb1,,bq;T).

Invariance

Laplace Transform

35.8.12 Ωetr(-TX)|X|γ-12(m+1)Fqp(a1,,apb1,,bq;-X)X=Γm(γ)|T|-γFqp+1(a1,,ap,γb1,,bq;-T-1),
(γ)>12(m-1).

Euler Integral

35.8.13 0<X<I|X|a1-12(m+1)|I-X|b1-a1-12(m+1)Fqp(a2,,ap+1b2,,bq+1;TX)X=1Bm(b1-a1,a1)Fq+1p+1(a1,,ap+1b1,,bq+1;T),
(b1-a1),(a1)>12(m-1).

§35.8(v) Mellin–Barnes Integrals

Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions Fqp and Fpp+1 of matrix argument. A similar result for the F10 function of matrix argument is given in Faraut and Korányi (1994, p. 346). These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case m=1.

See also Faraut and Korányi (1994, pp. 318–340).