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17 q-Hypergeometric and Related FunctionsProperties

§17.11 Transformations of q-Appell Functions

17.11.1 Φ(1)(a;b,b;c;x,y)=(a,bx,by;q)(c,x,y;q)ϕ23(c/a,x,ybx,by;q,a),
17.11.2 Φ(2)(a;b,b;c,c;x,y)=(b,ax;q)(c,x;q)n,r0(a,b;q)n(c/b,x;q)rbryn(q,c;q)n(q)r(ax;q)n+r,
17.11.3 Φ(3)(a,a;b,b;c;x,y)=(a,bx;q)(c,x;q)n,r0(a,b;q)n(x;q)r(c/a;q)n+raryn(q,c/a;q)n(q,bx;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 m1,,mn0(a;q)m1+m2++mn(b1;q)m1(b2;q)m2(bn;q)mnx1m1x2m2xnmn(q;q)m1(q;q)m2(q;q)mn(c;q)m1+m2++mn=(a,b1x1,b2x2,,bnxn;q)(c,x1,x2,,xn;q)ϕnn+1(c/a,x1,x2,,xnb1x1,b2x2,,bnxn;q,a).