Digital Library of Mathematical Functions
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17 q-Hypergeometric and Related FunctionsProperties

§17.3 q-Elementary and q-Special Functions


§17.3(i) Elementary Functions

q-Exponential Functions

17.3.1 eq(x)=n=0(1-q)nxn(q;q)n=1((1-q)x;q),
17.3.2 Eq(x)=n=0(1-q)nq(n2)xn(q;q)n=(-(1-q)x;q).

q-Sine Functions

17.3.3 sinq(x)=12(eq(x)-eq(-x))=n=0(1-q)2n+1(-1)nx2n+1(q;q)2n+1,
17.3.4 Sinq(x)=12(Eq(x)-Eq(-x))=n=0(1-q)2n+1qn(2n+1)(-1)nx2n+1(q;q)2n+1.

q-Cosine Functions

17.3.5 cosq(x)=12(eq(x)+eq(-x))=n=0(1-q)2n(-1)nx2n(q;q)2n,
17.3.6 Cosq(x)=12(Eq(x)+Eq(-x))=n=0(1-q)2nqn(2n-1)(-1)nx2n(q;q)2n.

See also Suslov (2003).

§17.3(ii) Gamma and Beta Functions

See §5.18.

§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers

q-Bernoulli Polynomials

17.3.7 βn(x,q)=(1-q)1-nr=0n(-1)r(nr)r+1(1-qr+1)qrx.

q-Euler Numbers

17.3.8 Am,s(q)=q(s-m2)+(s2)j=0s(-1)jq(j2)[m+1j]q(1-qs-j)m(1-q)m.

q-Stirling Numbers

17.3.9 am,s(q)=q-(s2)(1-q)s(q;q)sj=0s(-1)jq(j2)[sj]q(1-qs-j)m(1-q)m.

These were introduced in Carlitz (1954b, 1958). The βn(x,q) are, in fact, rational functions of q, and not necessarily polynomials. The Am,s(q) are always polynomials in q, and the am,s(q) are polynomials in q for 0sm.

§17.3(iv) Theta Functions

See §§17.8 and 20.5.

§17.3(v) Orthogonal Polynomials

See §§18.2718.29.