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17 q-Hypergeometric and Related FunctionsProperties

§17.10 Transformations of ψrr Functions

Bailey’s ψ22 Transformations

17.10.1 ψ22⁑(a,bc,d;q,z)=(a⁒z,d/a,c/b,d⁒q/(a⁒b⁒z);q)∞(z,d,q/b,c⁒d/(a⁒b⁒z);q)∞⁒ψ22⁑(a,a⁒b⁒z/da⁒z,c;q,da),
17.10.2 ψ22⁑(a,bc,d;q,z)=(a⁒z,b⁒z,c⁒q/(a⁒b⁒z),d⁒q/(a⁒b⁒z);q)∞(q/a,q/b,c,d;q)βˆžΓ—Οˆ22⁑(a⁒b⁒z/c,a⁒b⁒z/da⁒z,b⁒z;q,c⁒da⁒b⁒z).

Other Transformations

17.10.3 ψ88⁑(q⁒a12,βˆ’q⁒a12,c,d,e,f,a⁒qβˆ’n,qβˆ’na12,βˆ’a12,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f,qn+1,a⁒qn+1;q,a2⁒q2⁒n+2c⁒d⁒e⁒f)=(a⁒q,q/a,a⁒q/(c⁒d),a⁒q/(e⁒f);q)n(q/c,q/d,a⁒q/e,a⁒q/f;q)n⁒ψ44⁑(e,f,a⁒qn+1/(c⁒d),qβˆ’na⁒q/c,a⁒q/d,qn+1,e⁒f/(a⁒qn);q,q),
17.10.4 ψ22⁑(e,fa⁒q/c,a⁒q/d;q,a⁒qe⁒f)=(q/c,q/d,a⁒q/e,a⁒q/f;q)∞(a⁒q,q/a,a⁒q/(c⁒d),a⁒q/(e⁒f);q)βˆžΓ—βˆ‘n=βˆ’βˆžβˆž(1βˆ’a⁒q2⁒n)⁒(c,d,e,f;q)n(1βˆ’a)⁒(a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f;q)n⁒(q⁒a3c⁒d⁒e⁒f)n⁒qn2.
17.10.5 (a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,q/(a⁒b),q/(a⁒c),q/(a⁒d),q/(a⁒e);q)∞(f⁒a,g⁒a,f/a,g/a,q⁒a2,q/a2;q)βˆžΓ—Οˆ88⁑(q⁒a,βˆ’q⁒a,b⁒a,c⁒a,d⁒a,e⁒a,f⁒a,g⁒aa,βˆ’a,a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f,a⁒q/g;q,q2b⁒c⁒d⁒e⁒f⁒g)=(q,q/(b⁒f),q/(c⁒f),q/(d⁒f),q/(e⁒f),q⁒f/b,q⁒f/c,q⁒f/d,q⁒f/e;q)∞(f⁒a,q/(f⁒a),a⁒q/f,f/a,g/f,f⁒g,q⁒f2;q)βˆžΓ—Ο•78⁑(f2,q⁒f,βˆ’q⁒f,f⁒b,f⁒c,f⁒d,f⁒e,f⁒gf,βˆ’f,f⁒q/b,f⁒q/c,f⁒q/d,f⁒q/e,f⁒q/g;q,q2b⁒c⁒d⁒e⁒f⁒g)+idem⁑(f;g).
17.10.6 (a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f,q/(a⁒b),q/(a⁒c),q/(a⁒d),q/(a⁒e),q/(a⁒f);q)∞(a⁒g,a⁒h,a⁒k,g/a,h/a,k/a,q⁒a2,q/a2;q)βˆžΓ—Οˆ1010⁑(q⁒a,βˆ’q⁒a,b⁒a,c⁒a,d⁒a,e⁒a,f⁒a,g⁒a,h⁒a,k⁒aa,βˆ’a,a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f,a⁒q/g,a⁒q/h,a⁒q/k;q,q2b⁒c⁒d⁒e⁒f⁒g⁒h⁒k)=(q,q/(b⁒g),q/(c⁒g),q/(d⁒g),q/(e⁒g),q/(f⁒g),q⁒g/b,q⁒g/c,q⁒g/d,q⁒g/e,q⁒g/f;q)∞(g⁒h,g⁒k,h/g,a⁒g,q/(a⁒g),g/a,a⁒q/g,q⁒g2;q)βˆžΓ—Ο•910⁑(g2,q⁒g,βˆ’q⁒g,g⁒b,g⁒c,g⁒d,g⁒e,g⁒f,g⁒h,g⁒kg,βˆ’g,q⁒g/b,q⁒g/c,q⁒g/d,q⁒g/e,q⁒g/f,q⁒g/h,q⁒g/k;q,q2b⁒c⁒d⁒e⁒f⁒g⁒h⁒k)+idem⁑(g;h,k).