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17 q-Hypergeometric and Related FunctionsProperties

Β§17.3 q-Elementary and q-Special Functions

Contents
  1. Β§17.3(i) Elementary Functions
  2. Β§17.3(ii) Gamma and Beta Functions
  3. Β§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers
  4. Β§17.3(iv) Theta Functions
  5. Β§17.3(v) Orthogonal Polynomials

Β§17.3(i) Elementary Functions

q-Exponential Functions

17.3.1 eq⁑(x)=βˆ‘n=0∞(1βˆ’q)n⁒xn(q;q)n=1((1βˆ’q)⁒x;q)∞,
17.3.2 Eq⁑(x)=βˆ‘n=0∞(1βˆ’q)n⁒q(n2)⁒xn(q;q)n=(βˆ’(1βˆ’q)⁒x;q)∞.

q-Sine Functions

17.3.3 sinq⁑(x)=12⁒i⁒(eq⁑(i⁒x)βˆ’eq⁑(βˆ’i⁒x))=βˆ‘n=0∞(1βˆ’q)2⁒n+1⁒(βˆ’1)n⁒x2⁒n+1(q;q)2⁒n+1,
17.3.4 Sinq⁑(x)=12⁒i⁒(Eq⁑(i⁒x)βˆ’Eq⁑(βˆ’i⁒x))=βˆ‘n=0∞(1βˆ’q)2⁒n+1⁒qn⁒(2⁒n+1)⁒(βˆ’1)n⁒x2⁒n+1(q;q)2⁒n+1.

q-Cosine Functions

17.3.5 cosq⁑(x)=12⁒(eq⁑(i⁒x)+eq⁑(βˆ’i⁒x))=βˆ‘n=0∞(1βˆ’q)2⁒n⁒(βˆ’1)n⁒x2⁒n(q;q)2⁒n,
17.3.6 Cosq⁑(x)=12⁒(Eq⁑(i⁒x)+Eq⁑(βˆ’i⁒x))=βˆ‘n=0∞(1βˆ’q)2⁒n⁒qn⁒(2⁒nβˆ’1)⁒(βˆ’1)n⁒x2⁒n(q;q)2⁒n.

See also Suslov (2003).

Β§17.3(ii) Gamma and Beta Functions

See Β§5.18.

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

q-Bernoulli Polynomials

17.3.7 Ξ²n⁑(x,q)=(1βˆ’q)1βˆ’nβ’βˆ‘r=0n(βˆ’1)r⁒(nr)⁒r+1(1βˆ’qr+1)⁒qr⁒x.

q-Euler Numbers

17.3.8 Am,s⁑(q)=q(sβˆ’m2)+(s2)β’βˆ‘j=0s(βˆ’1)j⁒q(j2)⁒[m+1j]q⁒(1βˆ’qsβˆ’j)m(1βˆ’q)m.

q-Stirling Numbers

17.3.9 am,s⁑(q)=qβˆ’(s2)⁒(1βˆ’q)s(q;q)sβ’βˆ‘j=0s(βˆ’1)j⁒q(j2)⁒[sj]q⁒(1βˆ’qsβˆ’j)m(1βˆ’q)m.

These were introduced in Carlitz (1954a, 1958). The Ξ²n⁑(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 0≀s≀m.

Β§17.3(iv) Theta Functions

See §§17.8 and 20.5.

Β§17.3(v) Orthogonal Polynomials

See §§18.27–18.29.