Recursion formulae for basic hypergeometric functions

We show that the basic hypergeometric functionsFk(ω) ≔ ∏i=1r(ai;q)k(q;q)k∏j=1s(bj;q)kωk(−1)kq(k2)s+1−r r+1φs+1a1qk,…,arqk,αqk+1b1qk,…,bsqk,αβq2k+2q;ωqk(s+1−r),satisfy a recurrence relation of the form∑i=0ϑAi(k)+1ωBi(k)Fk+i(ω)=0, ϑ=max(r+1, s+2),where Ai(k), Bi(k) are rational functions of qk, and B0(k)=Bϑ(k)≡0. =s+1 and ω=q, this result can be refined. Namely, we show that the functionsFk(q) ≔ ∏i=1s+1(ai;q)k(q;q)k∏j=1s(bj;q)kqk s+2φs+1a1qk,…,as+1qk,αqk+1b1qk,…,bsqk,αβq2k+2q;q,satisfy a recurrence relation of order s+1,∑i=0s+1Ci(k)Fk+i(q)=0with rational coefficients in qk.