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17 q-Hypergeometric and Related FunctionsProperties

§17.2 Calculus

Contents

§17.2(i) q-Calculus

For n=0,1,2,,

17.2.1 (a;q)n=(1-a)(1-aq)(1-aqn-1),
17.2.2 (a;q)-n=1(aq-n;q)n=(-q/a)nq(n2)(q/a;q)n.

For ν

17.2.3 (a;q)ν=j=0(1-aqj1-aqν+j),

when this product converges.

17.2.4 (a;q) =j=0(1-aqj),
17.2.5 (a1,a2,,ar;q)n =j=1r(aj;q)n,
17.2.6 (a1,a2,,ar;q) =j=1r(aj;q).
17.2.7 (a;q-1)n=(a-1;q)n(-a)nq-(n2),
17.2.8 (a;q-1)n(b;q-1)n=(a-1;q)n(b-1;q)n(ab)n,
17.2.9 (a;q)n=(q1-n/a;q)n(-a)nq(n2),
17.2.10 (a;q)n(b;q)n=(q1-n/a;q)n(q1-n/b;q)n(ab)n,
17.2.11 (aq-n;q)n=(q/a;q)n(-aq)nq-(n2),
17.2.12 (aq-n;q)n(bq-n;q)n=(q/a;q)n(q/b;q)n(ab)n.
17.2.13 (a;q)n-k=(a;q)n(q1-n/a;q)k(-qa)kq(k2)-nk,
17.2.14 (a;q)n-k(b;q)n-k=(a;q)n(b;q)n(q1-n/b;q)k(q1-n/a;q)k(ba)k,
17.2.15 (aq-n;q)k=(a;q)k(q/a;q)n(q1-k/a;q)nq-nk,
17.2.16 (aq-n;q)n-k=(q/a;q)n(q/a;q)k(-aq)n-kq(k2)-(n2),
17.2.17 (aqn;q)k =(a;q)k(aqk;q)n(a;q)n,
17.2.18 (aqk;q)n-k =(a;q)n(a;q)k.
17.2.19 (a;q)2n=(a,aq;q2)n,

more generally,

17.2.20 (a;q)kn=(a,aq,,aqk-1;qk)n.
17.2.21 (a2;q2)n=(a;q)n(-a;q)n,
17.2.22 (qa12,-aq12;q)n(a12,-a12;q)n=(aq2;q2)n(a;q2)n=1-aq2n1-a,

more generally,

17.2.23 (aq1k,qωka1k,,qωkk-1a1k;q)n(a1k,ωka1k,,ωkk-1a1k;q)n=(aqk;qk)n(a;qk)n=1-aqkn1-a,

where ωk=e2πi/k.

17.2.24 limτ0(a/τ;q)nτn=limσ(aσ;q)nσ-n=(-a)nq(n2),
17.2.25 limτ0(a/τ;q)n(b/τ;q)n=limσ(aσ;q)n(bσ;q)n=(ab)n,
17.2.26 limτ0(a/τ;q)n(b/τ;q)n(c/τ2;q)n=(-1)n(abc)nq(n2).

§17.2(ii) Binomial Coefficients

17.2.27 [nm]q=(q;q)n(q;q)m(q;q)n-m=(q-n;q)m(-1)mqnm-(m2)(q;q)m,
17.2.28 limq1[nm]q=(nm)=n!m!(n-m)!,
17.2.29 [m+nm]q=(qn+1;q)m(q;q)m,
17.2.30 [-nm]q =[m+n-1m]q(-1)mq-mn-(m2),
17.2.31 [nm]q =[n-1m-1]q+qm[n-1m]q,
17.2.32 [nm]q =[n-1m]q+qn-m[n-1m-1]q,
17.2.33 limn[nm]q=1(q;q)m=1(1-q)(1-q2)(1-qm),
17.2.34 limn[rn+usn+t]q=1(q;q)=j=11(1-qj),

provided that r>s.

§17.2(iii) Binomial Theorem

In the limit as q1, (17.2.35) reduces to the standard binomial theorem

17.2.36 j=0n(nj)(-z)j=(1-z)n.

Also,

17.2.37 n=0(a;q)n(q;q)nzn=(az;q)(z;q),

provided that |z|<1. When a=qm+1, where m is a nonnegative integer, (17.2.37) reduces to the q-binomial series

17.2.38 n=0[n+mn]qzn =1(z;q)m+1.
17.2.39 j=0n[nj]q2qj =(-q;q)n,
17.2.40 j=02n(-1)j[2nj]q =(q;q2)n.

When n in (17.2.35), and when m in (17.2.38), the results become convergent infinite series and infinite products (see (17.5.1) and (17.5.4)).

See also §26.9(ii).

§17.2(iv) Derivatives

The q-derivatives of f(z) are defined by

17.2.41 𝒟qf(z)={f(z)-f(zq)(1-q)z,z0,f(0),z=0,

and

17.2.42 f[n](z)=𝒟qnf(z)={z-n(1-q)-nj=0nq-nj+(j+12)(-1)j[nj]qf(zqj),z0,f(n)(0)(q;q)nn!(1-q)n,z=0.

When q1 the q-derivatives converge to the corresponding ordinary derivatives.

Product Rule

17.2.43 𝒟q(f(z)g(z))=g(z)f[1](z)+f(zq)g[1](z).

Leibniz Rule

17.2.44 𝒟qn(f(z)g(z))=j=0n[nj]qf[n-j](zqj)g[j](z).

q-differential equations are considered in §17.6(iv).

§17.2(v) Integrals

If f(x) is continuous at x=0, then

17.2.45 01f(x)dqx=(1-q)j=0f(qj)qj,

and more generally,

17.2.46 0af(x)dqx=a(1-q)j=0f(aqj)qj.

If f(x) is continuous on [0,a], then

17.2.47 limq1-0af(x)dqx=0af(x)dx.

Infinite Range

17.2.48 0f(x)dqx=limn0q-nf(x)dqx=(1-q)j=-f(qj)qj,

provided that j=-f(qj)qj converges.

§17.2(vi) Rogers–Ramanujan Identities

17.2.49 1+n=1qn2(1-q)(1-q2)(1-qn)=n=01(1-q5n+1)(1-q5n+4),
17.2.50 1+n=1qn2+n(1-q)(1-q2)(1-qn)=n=01(1-q5n+2)(1-q5n+3).

These identities are the first in a large collection of similar results. See §17.14.