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17 q-Hypergeometric and Related FunctionsProperties

Β§17.2 Calculus

Contents
  1. Β§17.2(i) q-Calculus
  2. Β§17.2(ii) Binomial Coefficients
  3. Β§17.2(iii) Binomial Theorem
  4. Β§17.2(iv) Derivatives
  5. Β§17.2(v) Integrals
  6. Β§17.2(vi) Rogers–Ramanujan Identities

Β§17.2(i) q-Calculus

For n=0,1,2,…,

17.2.1 (a;q)n=(1βˆ’a)⁒(1βˆ’a⁒q)⁒⋯⁒(1βˆ’a⁒qnβˆ’1),
17.2.2 (a;q)βˆ’n=1(a⁒qβˆ’n;q)n=(βˆ’q/a)n⁒q(n2)(q/a;q)n.

For Ξ½βˆˆβ„‚

17.2.3 (a;q)Ξ½=∏j=0∞(1βˆ’a⁒qj1βˆ’a⁒qΞ½+j),

when this product converges.

17.2.4 (a;q)∞ =∏j=0∞(1βˆ’a⁒qj),
17.2.5 (a1,a2,…,ar;q)n =∏j=1r(aj;q)n,
17.2.6 (a1,a2,…,ar;q)∞ =∏j=1r(aj;q)∞.
For properties of the function f⁑(q)=qβˆ’1/24⁒η⁑(ln⁑q2⁒π⁒i)=(q;q)∞ see Β§27.14. Let q=eβˆ’t and q^=eβˆ’4⁒π2/t. Then
17.2.6_1 (q;q)∞ =2⁒πt⁒exp⁑(βˆ’Ο€26⁒t+t24)⁒(q^;q^)∞,
β„œβ‘t>0,
17.2.6_2 (βˆ’q;q)∞ =12⁒exp⁑(Ο€212⁒t+t24)⁒(q^12;q^)∞,
t>0.
For these and similar results see (Apostol, 1990, Ch.Β 3) and (Katsurada, 2003, Β§3). Note that (17.2.6_1) is just (27.14.14) with a=d=0 and βˆ’b=c=1.
17.2.7 (a;qβˆ’1)n=(aβˆ’1;q)n⁒(βˆ’a)n⁒qβˆ’(n2),
17.2.8 (a;qβˆ’1)n(b;qβˆ’1)n=(aβˆ’1;q)n(bβˆ’1;q)n⁒(ab)n,
17.2.9 (a;q)n=(q1βˆ’n/a;q)n⁒(βˆ’a)n⁒q(n2),
17.2.10 (a;q)n(b;q)n=(q1βˆ’n/a;q)n(q1βˆ’n/b;q)n⁒(ab)n,
17.2.11 (a⁒qβˆ’n;q)n=(q/a;q)n⁒(βˆ’aq)n⁒qβˆ’(n2),
17.2.12 (a⁒qβˆ’n;q)n(b⁒qβˆ’n;q)n=(q/a;q)n(q/b;q)n⁒(ab)n.
17.2.13 (a;q)nβˆ’k=(a;q)n(q1βˆ’n/a;q)k⁒(βˆ’qa)k⁒q(k2)βˆ’n⁒k,
17.2.14 (a;q)nβˆ’k(b;q)nβˆ’k=(a;q)n(b;q)n⁒(q1βˆ’n/b;q)k(q1βˆ’n/a;q)k⁒(ba)k,
17.2.15 (a⁒qβˆ’n;q)k=(a;q)k⁒(q/a;q)n(q1βˆ’k/a;q)n⁒qβˆ’n⁒k,
17.2.16 (a⁒qβˆ’n;q)nβˆ’k=(q/a;q)n(q/a;q)k⁒(βˆ’aq)nβˆ’k⁒q(k2)βˆ’(n2),
17.2.17 (a⁒qn;q)k =(a;q)k⁒(a⁒qk;q)n(a;q)n,
17.2.18 (a⁒qk;q)nβˆ’k =(a;q)n(a;q)k.
17.2.19 (a;q)2⁒n=(a,a⁒q;q2)n,

more generally,

17.2.20 (a;q)k⁒n=(a,a⁒q,…,a⁒qkβˆ’1;qk)n.
17.2.21 (a2;q2)n=(a;q)n⁒(βˆ’a;q)n,
17.2.22 (q⁒a12,βˆ’q⁒a12;q)n(a12,βˆ’a12;q)n=(a⁒q2;q2)n(a;q2)n=1βˆ’a⁒q2⁒n1βˆ’a,

more generally,

17.2.23 (q⁒a1k,q⁒ωk⁒a1k,…,q⁒ωkkβˆ’1⁒a1k;q)n(a1k,Ο‰k⁒a1k,…,Ο‰kkβˆ’1⁒a1k;q)n=(a⁒qk;qk)n(a;qk)n=1βˆ’a⁒qk⁒n1βˆ’a,

where Ο‰k=e2⁒π⁒i/k.

17.2.24 limΟ„β†’0⁑(a/Ο„;q)n⁒τn=limΟƒβ†’βˆžβ‘(a⁒σ;q)nβ’Οƒβˆ’n=(βˆ’a)n⁒q(n2),
17.2.25 limΟ„β†’0⁑(a/Ο„;q)n(b/Ο„;q)n=limΟƒβ†’βˆžβ‘(a⁒σ;q)n(b⁒σ;q)n=(ab)n,
17.2.26 limΟ„β†’0⁑(a/Ο„;q)n⁒(b/Ο„;q)n(c/Ο„2;q)n=(βˆ’1)n⁒(a⁒bc)n⁒q(n2).

Β§17.2(ii) Binomial Coefficients

17.2.27 [nm]q=(q;q)n(q;q)m⁒(q;q)nβˆ’m=(qβˆ’n;q)m⁒(βˆ’1)m⁒qn⁒mβˆ’(m2)(q;q)m,
17.2.28 limqβ†’1⁑[nm]q=(nm)=n!m!⁒(nβˆ’m)!,
17.2.29 [m+nm]q=(qn+1;q)m(q;q)m,
17.2.30 [βˆ’nm]q =[m+nβˆ’1m]q⁒(βˆ’1)m⁒qβˆ’m⁒nβˆ’(m2),
17.2.31 [nm]q =[nβˆ’1mβˆ’1]q+qm⁒[nβˆ’1m]q,
17.2.32 [nm]q =[nβˆ’1m]q+qnβˆ’m⁒[nβˆ’1mβˆ’1]q,
17.2.33 limnβ†’βˆžβ‘[nm]q=1(q;q)m=1(1βˆ’q)⁒(1βˆ’q2)⁒⋯⁒(1βˆ’qm),
17.2.34 limnβ†’βˆžβ‘[r⁒n+us⁒n+t]q=1(q;q)∞=∏j=1∞1(1βˆ’qj),

provided that r>s.

Β§17.2(iii) Binomial Theorem

17.2.35 βˆ‘j=0n[nj]q⁒(βˆ’z)j⁒q(j2)=(z;q)n=(1βˆ’z)⁒(1βˆ’z⁒q)⁒⋯⁒(1βˆ’z⁒qnβˆ’1).

In the limit as q→1, (17.2.35) reduces to the standard binomial theorem

17.2.36 βˆ‘j=0n(nj)⁒(βˆ’z)j=(1βˆ’z)n.

Also,

17.2.37 βˆ‘n=0∞(a;q)n(q;q)n⁒zn=(a⁒z;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 βˆ‘n=0∞[n+mn]q⁒zn =1(z;q)m+1.
17.2.39 βˆ‘j=0n[nj]q2⁒qj =(βˆ’q;q)n,
17.2.40 βˆ‘j=02⁒n(βˆ’1)j⁒[2⁒nj]q =(q;q2)n.

When nβ†’βˆž in (17.2.35), and when mβ†’βˆž in (17.2.38), the results become convergent infinite series and infinite products (see (17.5.1) and (17.5.4)).

See also Β§26.9(ii).

Β§17.2(iv) Derivatives

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

17.2.41 π’Ÿq⁑f⁑(z)={f⁑(z)βˆ’f⁑(z⁒q)(1βˆ’q)⁒z,zβ‰ 0,f′⁑(0),z=0,

and

17.2.42 f[n]⁑(z)=π’Ÿqn⁑f⁑(z)={zβˆ’n⁒(1βˆ’q)βˆ’nβ’βˆ‘j=0nqβˆ’n⁒j+(j+12)⁒(βˆ’1)j⁒[nj]q⁒f⁑(z⁒qj),zβ‰ 0,f(n)⁑(0)⁒(q;q)nn!⁒(1βˆ’q)n,z=0.

When q→1 the q-derivatives converge to the corresponding ordinary derivatives.

Product Rule

17.2.43 π’Ÿq⁑(f⁑(z)⁒g⁑(z))=g⁑(z)⁒f[1]⁑(z)+f⁑(z⁒q)⁒g[1]⁑(z).

Leibniz Rule

17.2.44 π’Ÿqn⁑(f⁑(z)⁒g⁑(z))=βˆ‘j=0n[nj]q⁒f[nβˆ’j]⁑(z⁒qj)⁒g[j]⁑(z).

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

Β§17.2(v) Integrals

If f⁑(x) is continuous at x=0, then

17.2.45 ∫01f⁑(x)⁒dqx=(1βˆ’q)β’βˆ‘j=0∞f⁑(qj)⁒qj,

and more generally,

17.2.46 ∫0af⁑(x)⁒dqx=a⁒(1βˆ’q)β’βˆ‘j=0∞f⁑(a⁒qj)⁒qj.

If f⁑(x) is continuous on [0,a], then

17.2.47 limqβ†’1βˆ’β‘βˆ«0af⁑(x)⁒dqx=∫0af⁑(x)⁒dx.

Infinite Range

17.2.48 ∫0∞f⁑(x)⁒dqx=limnβ†’βˆžβ‘βˆ«0qβˆ’nf⁑(x)⁒dqx=(1βˆ’q)β’βˆ‘j=βˆ’βˆžβˆžf⁑(qj)⁒qj,

provided that βˆ‘j=βˆ’βˆžβˆžf⁑(qj)⁒qj converges.

Β§17.2(vi) Rogers–Ramanujan Identities

17.2.49 1+βˆ‘n=1∞qn2(1βˆ’q)⁒(1βˆ’q2)⁒⋯⁒(1βˆ’qn)=∏n=0∞1(1βˆ’q5⁒n+1)⁒(1βˆ’q5⁒n+4),
17.2.50 1+βˆ‘n=1∞qn2+n(1βˆ’q)⁒(1βˆ’q2)⁒⋯⁒(1βˆ’qn)=∏n=0∞1(1βˆ’q5⁒n+2)⁒(1βˆ’q5⁒n+3).

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