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16 Generalized Hypergeometric Functions and Meijer G-FunctionGeneralized Hypergeometric Functions

§16.4 Argument Unity

Contents

§16.4(i) Classification

The function Fqq+1(a;b;z) is well-poised if

16.4.1 a1+b1==aq+bq=aq+1+1.

It is very well-poised if it is well-poised and a1=b1+1.

The special case Fqq+1(a;b;1) is k-balanced if aq+1 is a nonpositive integer and

16.4.2 a1++aq+1+k=b1++bq.

When k=1 the function is said to be balanced or Saalschützian.

§16.4(ii) Examples

The function Fqq+1 with argument unity and general values of the parameters is discussed in Bühring (1992). Special cases are as follows:

Pfaff–Saalschütz Balanced Sum

16.4.3 F23(-n,a,bc,d;1)=(c-a)n(c-b)n(c)n(c-a-b)n,

when c+d=a+b+1-n, n=0,1,. See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity.

For generalizations involving Fr+2r+3 functions see Kim et al. (2013).

Dixon’s Well-Poised Sum

16.4.4 F23(a,b,ca-b+1,a-c+1;1)=Γ(12a+1)Γ(a-b+1)Γ(a-c+1)Γ(12a-b-c+1)Γ(a+1)Γ(12a-b+1)Γ(12a-c+1)Γ(a-b-c+1),

when (a-2b-2c)>-2, or when the series terminates with a=-n:

16.4.5 F23(-n,b,c1-b-n,1-c-n;1)={0,n=2k+1,(2k)!Γ(b+k)Γ(c+k)Γ(b+c+2k)k!Γ(b+2k)Γ(c+2k)Γ(b+c+k),n=2k,

where k=0,1,.

Watson’s Sum

16.4.6 F23(a,b,c12(a+b+1),2c;1)=Γ(12)Γ(c+12)Γ(12(a+b+1))Γ(c+12(1-a-b))Γ(12(a+1))Γ(12(b+1))Γ(c+12(1-a))Γ(c+12(1-b)),

when (2c-a-b)>-1, or when the series terminates with a=-n.

Whipple’s Sum

16.4.7 F23(a,1-a,cd,2c-d+1;1)=πΓ(d)Γ(2c-d+1)21-2cΓ(c+12(a-d+1))Γ(c+1-12(a+d))Γ(12(a+d))Γ(12(d-a+1)),

when c>0 or when a is an integer.

Džrbasjan’s Sum

This is (16.4.7) in the case c=-n:

16.4.8 F23(-n,a,1-ad,1-d-2n;1)=(12(a+d))n(12(d-a+1))n(12d)n(12(d+1))n,
n=0,1,.

Rogers–Dougall Very Well-Poised Sum

16.4.9 F45(a,12a+1,b,c,d12a,a-b+1,a-c+1,a-d+1;1)=Γ(a-b+1)Γ(a-c+1)Γ(a-d+1)Γ(a-b-c-d+1)Γ(a+1)Γ(a-b-c+1)Γ(a-b-d+1)Γ(a-c-d+1),

when (b+c+d-a)<1, or when the series terminates with d=-n.

Dougall’s Very Well-Poised Sum

16.4.10 F67(a,12a+1,b,c,d,f,-n12a,a-b+1,a-c+1,a-d+1,a-f+1,a+n+1;1)=(a+1)n(a-b-c+1)n(a-b-d+1)n(a-c-d+1)n(a-b+1)n(a-c+1)n(a-d+1)n(a-b-c-d+1)n,
n=0,1,,

when 2a+1=b+c+d+f-n. The last condition is equivalent to the sum of the top parameters plus 2 equals the sum of the bottom parameters, that is, the series is 2-balanced.

§16.4(iii) Identities

16.4.11 F23(a,b,cd,e;1)=Γ(e)Γ(d+e-a-b-c)Γ(e-a)Γ(d+e-b-c)F23(a,d-b,d-cd,d+e-b-c;1),

when (d+e-a-b-c)>0 and (e-a)>0. The function F23(a,b,c;d,e;1) is analytic in the parameters a,b,c,d,e when its series expansion converges and the bottom parameters are not negative integers or zero. (16.4.11) provides a partial analytic continuation to the region when the only restrictions on the parameters are (e-a)>0, and d,e, and d+e-b-c0,-1,. A detailed treatment of analytic continuation in (16.4.11) and asymptotic approximations as the variables a,b,c,d,e approach infinity is given by Aomoto (1987).

There are two types of three-term identities for F23’s. The first are recurrence relations that extend those for F12’s; see §15.5(ii). Examples are (16.3.7) with z=1. Also,

16.4.12 (a-d)(b-d)(c-d)(F23(a,b,cd+1,e;1)-F23(a,b,cd,e;1))+abcF23(a,b,cd,e;1)=d(d-1)(a+b+c-d-e+1)(F23(a,b,cd,e;1)-F23(a,b,cd-1,e;1)),

and

16.4.13 F23(a,b,cd,e;1)=c(e-a)deF23(a,b+1,c+1d+1,e+1;1)+d-cdF23(a,b+1,cd+1,e;1).

Methods of deriving such identities are given by Bailey (1964), Rainville (1960), Raynal (1979), and Wilson (1978). Lists are given by Raynal (1979) and Wilson (1978). See Raynal (1979) for a statement in terms of 3j symbols (Chapter 34). Also see Wilf and Zeilberger (1992a, b) for information on the Wilf–Zeilberger algorithm which can be used to find such relations.

The other three-term relations are extensions of Kummer’s relations for F12’s given in §15.10(ii). See Bailey (1964, pp. 19–22).

Balanced F34(1) series have transformation formulas and three-term relations. The basic transformation is given by

16.4.14 F34(-n,a,b,cd,e,f;1)=(e-a)n(f-a)n(e)n(f)nF34(-n,a,d-b,d-cd,a-e-n+1,a-f-n+1;1),

when a+b+c-n+1=d+e+f. These series contain 6j symbols as special cases when the parameters are integers; compare §34.4.

The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs.

Contiguous balanced series have parameters shifted by an integer but still balanced. One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. See Raynal (1979), Wilson (1978), and Bailey (1964).

A different type of transformation is that of Whipple:

16.4.15 F67(a,12a+1,b,c,d,e,f12a,a-b+1,a-c+1,a-d+1,a-e+1,a-f+1;1)=Γ(a-d+1)Γ(a-e+1)Γ(a-f+1)Γ(a-d-e-f+1)Γ(a+1)Γ(a-d-e+1)Γ(a-d-f+1)Γ(a-e-f+1)F34(a-b-c+1,d,e,fa-b+1,a-c+1,d+e+f-a;1),

when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. See Bailey (1964, §4.4(4)).

Transformations for both balanced F34(1) and very well-poised F67(1) are included in Bailey (1964, pp. 56–63). A similar theory is available for very well-poised F89(1)’s which are 2-balanced. See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations.

Relations between three solutions of three-term recurrence relations are given by Masson (1991). See also Lewanowicz (1985) (with corrections in Lewanowicz (1987)) for further examples of recurrence relations.

§16.4(iv) Continued Fractions

For continued fractions for ratios of F23 functions with argument unity, see Cuyt et al. (2008, pp. 315–317).

§16.4(v) Bilateral Series

Denote, formally, the bilateral hypergeometric function

16.4.16 Hqp(a1,,apb1,,bq;z)=k=-(a1)k(ap)k(b1)k(bq)kzk.

Then

16.4.17 H22(a,bc,d;1)=Γ(c)Γ(d)Γ(1-a)Γ(1-b)Γ(c+d-a-b-1)Γ(c-a)Γ(d-a)Γ(c-b)Γ(d-b),
(c+d-a-b)>1.

This is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).