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35 Functions of Matrix ArgumentProperties

§35.8 Generalized Hypergeometric Functions of Matrix Argument

  1. §35.8(i) Definition
  2. §35.8(ii) Relations to Other Functions
  3. §35.8(iii) F23 Case
  4. §35.8(iv) General Properties
  5. §35.8(v) Mellin–Barnes Integrals

§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

35.8.2 F00(;𝐓)=etr(𝐓),
35.8.3 F12(a,bb;𝐓)=F01(a;𝐓)=|𝐈𝐓|a,
35.8.4 Aν(𝐓)=1Γm(ν+12(m+1))F10(ν+12(m+1);𝐓),

§35.8(iii) F23 Case

Kummer Transformation

Let c=b1+b2a1a2a3. Then

35.8.5 F23(a1,a2,a3b1,b2;𝐈)=Γm(b2)Γm(c)Γm(b2a3)Γm(c+a3)F23(b1a1,b1a2,a3b1,c+a3;𝐈),

Pfaff–Saalschütz Formula

Let a1+a2+a3+12(m+1)=b1+b2; one of the aj be a negative integer; (b1a1), (b1a2), (b1a3), (b1a1a2a3)>12(m1). Then

35.8.6 F23(a1,a2,a3b1,b2;𝐈)=Γm(b1a1)Γm(b1a2)Γm(b1)Γm(b1a1a2)Γm(b1a3)Γm(b1a1a2a3)Γm(b1a1a3)Γm(b1a2a3).

Thomae Transformation

Again, let c=b1+b2a1a2a3. 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(b1a1,b2a2,cc+a2,c+a3;𝐈),
(b1), (b2), (c)>12(m1).

§35.8(iv) General Properties

Value at 𝐓=𝟎



Laplace Transform

Euler Integral

§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).