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Β§17.7 Special Cases of Higher Ο•sr Functions

Contents
  1. Β§17.7(i) Ο•22 Functions
  2. Β§17.7(ii) Ο•23 Functions
  3. Β§17.7(iii) Other Ο•sr Functions

Β§17.7(i) Ο•22 Functions

q-Analog of Bailey’s F12⁑(βˆ’1) Sum

17.7.1 Ο•22⁑(a,q/aβˆ’q,b;q,βˆ’b)=(a⁒b,b⁒q/a;q2)∞(b;q)∞.

q-Analog of Gauss’s F12⁑(βˆ’1) Sum

17.7.2 Ο•22⁑(a2,b2a⁒b⁒q12,βˆ’a⁒b⁒q12;q,βˆ’q)=(a2⁒q,b2⁒q;q2)∞(q,a2⁒b2⁒q;q2)∞.

Sum Related to (17.6.4)

17.7.3 Ο•22⁑(c2/b2,b2c,c⁒q;q2,q)=12⁒(b2,q;q2)∞(c,c⁒q;q2)∞⁒((c/b;q)∞(b;q)∞+(βˆ’c/b;q)∞(βˆ’b;q)∞).

Β§17.7(ii) Ο•23 Functions

q-Pfaff–SaalschΓΌtz Sum

Nonterminating Form of the q-SaalschΓΌtz Sum

17.7.5 Ο•23⁑(a,b,ce,f;q,q)+(q/e,a,b,c,q⁒f/e;q)∞(e/q,a⁒q/e,b⁒q/e,c⁒q/e,f;q)βˆžβ’Ο•23⁑(a⁒q/e,b⁒q/e,c⁒q/eq2/e,q⁒f/e;q,q)=(q/e,f/a,f/b,f/c;q)∞(a⁒q/e,b⁒q/e,c⁒q/e,f;q)∞,

where e⁒f=a⁒b⁒c⁒q.

F.Β H.Β Jackson’s Terminating q-Analog of Dixon’s Sum

17.7.6 Ο•23⁑(qβˆ’2⁒n,b,cq1βˆ’2⁒n/b,q1βˆ’2⁒n/c;q,q2βˆ’nb⁒c)=(b,c;q)n⁒(q,b⁒c;q)2⁒n(q,b⁒c;q)n⁒(b,c;q)2⁒n.

Continued Fractions

For continued-fraction representations of a ratio of Ο•23 functions, see Cuyt et al. (2008, pp.Β 399–400).

Β§17.7(iii) Other Ο•sr Functions

q-Analog of Dixon’s F23⁑(1) Sum

17.7.7 Ο•34⁑(a,βˆ’q⁒a12,b,cβˆ’a12,a⁒q/b,a⁒q/c;q,q⁒a12b⁒c)=(a⁒q,q⁒a12/b,q⁒a12/c,a⁒q/(b⁒c);q)∞(a⁒q/b,a⁒q/c,q⁒a12,q⁒a12/(b⁒c);q)∞.

Gasper–Rahman q-Analog of Watson’s F23 Sum

17.7.8 Ο•78⁑(Ξ»,q⁒λ12,βˆ’q⁒λ12,a,b,c,βˆ’c,λ⁒q/c2Ξ»12,βˆ’Ξ»12,λ⁒q/a,λ⁒q/b,λ⁒q/c,βˆ’Ξ»β’q/c,c2;q,βˆ’Ξ»β’qa⁒b)=(λ⁒q,c2/Ξ»;q)∞⁒(a⁒q,b⁒q,c2⁒q/a,c2⁒q/b;q2)∞(λ⁒q/a,λ⁒q/b;q)∞⁒(q,a⁒b⁒q,c2⁒q,c2⁒q/(a⁒b);q2)∞,

where Ξ»=βˆ’c⁒(a⁒b/q)12.

Andrews’ q-Analog of the Terminating Version of Watson’s F23 Sum (16.4.6)

17.7.9 Ο•34⁑(qβˆ’n,a⁒qn,c,βˆ’c(a⁒q)12,βˆ’(a⁒q)12,c2;q,q)={0,nΒ odd,cn⁒(q,a⁒q/c2;q2)n/2(a⁒q,c2⁒q;q2)n/2,nΒ even.

Gasper–Rahman q-Analog of Whipple’s F23 Sum

17.7.10 Ο•78⁑(βˆ’c,q⁒(βˆ’c)12,βˆ’q⁒(βˆ’c)12,a,q/a,c,βˆ’d,βˆ’q/d(βˆ’c)12,βˆ’(βˆ’c)12,βˆ’c⁒q/a,βˆ’a⁒c,βˆ’q,c⁒q/d,c⁒d;q,c)=(βˆ’c,βˆ’c⁒q;q)∞⁒(a⁒c⁒d,a⁒c⁒q/d,c⁒d⁒q/a,c⁒q2/(a⁒d);q2)∞(c⁒d,c⁒q/d,βˆ’a⁒c,βˆ’c⁒q/a;q)∞.

Andrews’ q-Analog of the Terminating Version of Whipple’s F23 Sum (16.4.7)

17.7.11 Ο•34⁑(qβˆ’n,qn+1,c,βˆ’ce,c2⁒q/e,βˆ’q;q,q)=q(n+12)⁒(e⁒qβˆ’n,e⁒qn+1,c2⁒q1βˆ’n/e,c2⁒qn+2/e;q2)∞(e,c2⁒q/e;q)∞.

First q-Analog of Bailey’s F34⁑(1) Sum

17.7.12 Ο•34⁑(a,a⁒q,b2⁒q2⁒n,qβˆ’2⁒nb,b⁒q,a2⁒q2;q2,q2)=an⁒(βˆ’q,b/a;q)n(βˆ’a⁒q,b;q)n.

Second q-Analog of Bailey’s F34⁑(1) Sum

17.7.13 Ο•34⁑(a,a⁒q,b2⁒q2⁒nβˆ’2,qβˆ’2⁒nb,b⁒q,a2;q2,q2)=an⁒(βˆ’q,b/a;q)n⁒(1βˆ’b⁒qnβˆ’1)(βˆ’a,b;q)n⁒(1βˆ’b⁒q2⁒nβˆ’1).

F.Β H.Β Jackson’s q-Analog of Dougall’s F67⁑(1) Sum

17.7.14 Ο•78⁑(a,q⁒a12,βˆ’q⁒a12,b,c,d,e,qβˆ’na12,βˆ’a12,a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒qn+1;q,q)=(a⁒q,a⁒q/(b⁒c),a⁒q/(b⁒d),a⁒q/(c⁒d);q)n(a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/(b⁒c⁒d);q)n,

where a2⁒q=b⁒c⁒d⁒e⁒qβˆ’n.

Limiting Cases of (17.7.14)

17.7.15 Ο•56⁑(a,q⁒a12,βˆ’q⁒a12,b,c,da12,βˆ’a12,a⁒q/b,a⁒q/c,a⁒q/d;q,a⁒qb⁒c⁒d)=(a⁒q,a⁒q/(b⁒c),a⁒q/(b⁒d),a⁒q/(c⁒d);q)∞(a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/(b⁒c⁒d);q)∞,

and when d=qβˆ’n,

17.7.16 Ο•56⁑(a,q⁒a12,βˆ’q⁒a12,b,c,qβˆ’na12,βˆ’a12,a⁒q/b,a⁒q/c,a⁒qn+1;q,a⁒qn+1b⁒c)=(a⁒q,a⁒q/(b⁒c);q)n(a⁒q/b,a⁒q/c;q)n.

Bailey’s Nonterminating Extension of Jackson’s Ο•78 Sum

17.7.17 Ο•78⁑(a,q⁒a12,βˆ’q⁒a12,b,c,d,e,fa12,βˆ’a12,a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f;q,q)βˆ’ba⁒(a⁒q,c,d,e,f,b⁒q/a,b⁒q/c,b⁒q/d,b⁒q/e,b⁒q/f;q)∞(a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f,b⁒c/a,b⁒d/a,b⁒e/a,b⁒f/a,b2⁒q/a;q)βˆžΓ—Ο•78⁑(b2/a,q⁒b⁒aβˆ’12,βˆ’q⁒b⁒aβˆ’12,b,b⁒c/a,b⁒d/a,b⁒e/a,b⁒f/ab⁒aβˆ’12,βˆ’b⁒aβˆ’12,b⁒q/a,b⁒q/c,b⁒q/d,b⁒q/e,b⁒q/f;q,q)=(a⁒q,b/a,a⁒q/(c⁒d),a⁒q/(c⁒e),a⁒q/(c⁒f),a⁒q/(d⁒e),a⁒q/(d⁒f),a⁒q/(e⁒f);q)∞(a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f,b⁒c/a,b⁒d/a,b⁒e/a,b⁒f/a;q)∞,

where q⁒a2=b⁒c⁒d⁒e⁒f.

Gasper–Rahman q-Analogs of the Karlsson–Minton Sums

17.7.18 Ο•r+1r+2⁑(a,b,b1⁒qm1,…,br⁒qmrb⁒q,b1,…,br;q,aβˆ’1⁒q1βˆ’(m1+β‹―+mr))=(q,b⁒q/a;q)∞⁒(b1/b;q)m1⁒⋯⁒(br/b;q)mr(b⁒q,q/a;q)∞⁒(b1;q)m1⁒⋯⁒(br;q)mr⁒bm1+β‹―+mr,

and

17.7.19 Ο•rr+1⁑(a,b1⁒qm1,…,br⁒qmrb1,…,br;q,aβˆ’1⁒q1βˆ’(m1+β‹―+mr))=0,

where m1,m2,…,mr are arbitrary nonnegative integers.

Gosper’s Bibasic Sum

17.7.20 βˆ‘k=0n1βˆ’a⁒pk⁒qk1βˆ’a⁒(a;p)k⁒(c;q)k(q;q)k⁒(a⁒p/c;p)k⁒cβˆ’k=(a⁒p;p)n⁒(c⁒q;q)n(q;q)n⁒(a⁒p/c;p)n⁒cβˆ’n.

Gasper’s Extensions of Gosper’s Bibasic Sum

17.7.21 βˆ‘k=0n(1βˆ’a⁒pk⁒qk)⁒(1βˆ’b⁒pk⁒qβˆ’k)(1βˆ’a)⁒(1βˆ’b)⁒(a,b;p)k⁒(c,a/(b⁒c);q)k(q,a⁒q/b;q)k⁒(a⁒p/c,b⁒c⁒p;p)k⁒qk=(a⁒p,b⁒p;p)n⁒(c⁒q,a⁒q/(b⁒c);q)n(q,a⁒q/b;q)n⁒(a⁒p/c,b⁒c⁒p;p)n,
17.7.22 βˆ‘k=βˆ’mn(1βˆ’a⁒d⁒pk⁒qk)⁒(1βˆ’b⁒pk/(d⁒qk))(1βˆ’a⁒d)⁒(1βˆ’(b/d))Γ—(a,b;p)k⁒(c,a⁒d2/(b⁒c);q)k(d⁒q,a⁒d⁒q/b;q)k⁒(a⁒d⁒p/c,b⁒c⁒p/d;p)k⁒qk=(1βˆ’a)⁒(1βˆ’b)⁒(1βˆ’c)⁒(1βˆ’(a⁒d2/(b⁒c)))d⁒(1βˆ’a⁒d)⁒(1βˆ’(b/d))⁒(1βˆ’(c/d))⁒(1βˆ’(a⁒d/(b⁒c)))Γ—((a⁒p,b⁒p;p)n⁒(c⁒q,a⁒d2⁒q/(b⁒c);q)n(d⁒q,a⁒d⁒q/b;q)n⁒(a⁒d⁒p/c,b⁒c⁒p/d;p)nβˆ’(c/(a⁒d),d/(b⁒c);p)m+1⁒(1/d,b/(a⁒d);q)m+1(1/c,b⁒c/(a⁒d2);q)m+1⁒(1/a,1/b;p)m+1),

and n-th difference generalization:

17.7.23 (1βˆ’aq)⁒(1βˆ’bq)β’βˆ‘k=0n(a⁒pk,b⁒pβˆ’k;q)nβˆ’1⁒(1βˆ’(a⁒p2⁒k/b))(p;p)n⁒(p;p)nβˆ’k⁒(a⁒pk/b;q)n+1⁒(βˆ’1)k⁒p(k2)=Ξ΄n,0.