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17 q-Hypergeometric and Related FunctionsProperties

Β§17.11 Transformations of q-Appell Functions

17.11.1 Ξ¦(1)⁑(a;b,bβ€²;c;q;x,y)=(a,b⁒x,b′⁒y;q)∞(c,x,y;q)βˆžβ’Ο•23⁑(c/a,x,yb⁒x,b′⁒y;q,a),
17.11.2 Ξ¦(2)⁑(a;b,bβ€²;c,cβ€²;q;x,y)=(b,a⁒x;q)∞(c,x;q)βˆžβ’βˆ‘n,r≧0(a,bβ€²;q)n⁒(c/b,x;q)r⁒br⁒yn(q,cβ€²;q)n⁒(q;q)r⁒(a⁒x;q)n+r,
17.11.3 Ξ¦(3)⁑(a,aβ€²;b,bβ€²;c;q;x,y)=(a,b⁒x;q)∞(c,x;q)βˆžβ’βˆ‘n,r≧0(aβ€²,bβ€²;q)n⁒(x;q)r⁒(c/a;q)n+r⁒ar⁒yn(q,c/a;q)n⁒(q,b⁒x;q)r.

Of (17.11.1)–(17.11.3) only (17.11.1) has a natural generalization: the following sum reduces to (17.11.1) when n=2.

17.11.4 βˆ‘m1,…,mn≧0(a;q)m1+m2+β‹―+mn⁒(b1;q)m1⁒(b2;q)m2⁒⋯⁒(bn;q)mn⁒x1m1⁒x2m2⁒⋯⁒xnmn(q;q)m1⁒(q;q)m2⁒⋯⁒(q;q)mn⁒(c;q)m1+m2+β‹―+mn=(a,b1⁒x1,b2⁒x2,…,bn⁒xn;q)∞(c,x1,x2,…,xn;q)βˆžβ’Ο•nn+1⁑(c/a,x1,x2,…,xnb1⁒x1,b2⁒x2,…,bn⁒xn;q,a).