Index Q

$q$ -Appell functions §17.4(iii)
- transformations §17.11
$q$ -Askey scheme §18.27(i)
$q$ -Bernoulli polynomials §17.3(iii)
$q$ -beta function §5.18(iii)
$q$ -binomial coefficient §17.2(ii), §26.9(ii)
$q$ -binomial series §17.5
$q$ -binomial theorem §17.2(iii), §17.5
$q$ -calculus §17.2—§17.2(vi)
$q$ -commutator §18.38(iii)
$q$ -cosine function §17.3(i)
$q$ -derivatives §17.2(iv)
$q$ -differential equations §17.6(iv)
$q$ -Dyson conjecture §17.14
$q$ -elementary functions §17.17, §17.3(i)
$q$ -Euler numbers §17.3(iii)
$q$ -exponential function
- applications §17.17
$q$ -exponential function §17.3(i)
$q$ -factorials §5.18(i)
$q$ -gamma function §5.18(ii)
$q$ -Hahn class orthogonal polynomials §18.27—§18.27(vii)
- as eigenvalues of $q$ -difference operator §18.27(i)
- asymptotic approximations §18.29
- limit relations §18.27(iii), §18.27(vii), §18.27(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(vi)
- orthogonality properties §18.27(i)
- relation to $q$ -hypergeometric function §18.27(ii), §18.27(iii), §18.27(iv), §18.27(vii), §18.27(v), §18.27(vi), §18.27(vii)
$q$ -Hahn polynomials §18.27(ii)
$q$ -hypergeometric function §17.1
- Andrews–Askey sum §17.6(i), §17.7(i)
- Andrews’ $q$ -Dyson conjecture §17.14
- applications
  - mathematical §17.16
  - physical §17.17
- Bailey chain §17.12
- Bailey lemma
  - strong §17.12
  - weak §17.12
- Bailey pairs §17.12
- Bailey transform §17.12
- Bailey–Daum $q$ -Kummer sum §17.6(i)
- Bailey’s ${{}_{2}F_{1}}(-1)$ sum
  - $q$ -analog §17.7(i)
- Bailey’s ${{}_{4}F_{3}}(1)$ sum
  - $q$ -analogs (first and second) §17.7(iii)
- Bailey’s transformation of very-well-poised ${{}_{8}\phi_{7}}$ §17.9(iii)
- balanced series §17.4(iv)
- bibasic sums and series §17.7(iii), §17.9(iv)
- bilateral, see bilateral $q$ -hypergeometric function.
- Cauchy’s sum §17.5
- Chu–Vandermonde sums (first and second)
  - $q$ -analogs §17.6(i)
- computation §17.18
- constant term identities §17.14
- contiguous relations (Heine’s) §17.6(iii)
- continued fractions §17.6(vi), §17.7(ii)
- definition §17.4
- differential equations §17.6(iv)
- Dixon’s ${{}_{3}F_{2}}(1)$ sum
  - $q$ -analog §17.7(iii)
- Dixon’s sum
  - F. H. Jackson’s $q$ -analog §17.7(ii)
- Dougall’s ${{}_{7}F_{6}}(1)$ sum
  - F. H. Jackson’s $q$ -analog §17.7(iii)
- Euler’s sums (first, second, third) §17.5, §17.5
- F. H. Jackson’s transformations §17.9(i)
- Fine’s transformations (first, second, third) §17.6(ii)—§17.6(ii)
- Gauss’s ${{}_{2}F_{1}}(-1)$ sum
  - $q$ -analog §17.7(i)
- generalizations §17.15
- Heine’s transformations (first, second, third) §17.6(ii)
- idem function §17.1, §17.10
- integral representations §17.6(v)
- integrals §17.13
- $k$ -balanced series §17.4(iv)
- mixed base Heine-type transformations §17.9(iv)
- nearly-poised §17.4(iv)
- notation §17.1
- $q$ -Pfaff–Saalschütz sum §17.7(ii)
- $q$ -Saalschütz sum
  - nonterminating form §17.7(ii)
- $q$ -Sheppard identity §17.9(ii)
- quintuple product identity §17.8
- Ramanujan’s integrals §17.13
- relations to other functions
  - Askey–Wilson class orthogonal polynomials §18.28(ii)—§18.28(xi)
  - $q$ -Hahn class orthogonal polynomials §18.27—§18.27(vii)
- Rogers–Fine identity §17.6(ii)
- Saalschützian series §17.4(iv)
- Sears’ balanced ${{}_{4}\phi_{3}}$ transformation §17.9(iii)
- special cases §17.7
- three-term ${{}_{2}\phi_{1}}$ transformation §17.6(ii)
- transformations §17.9
- Vandermonde sum
  - nonterminating $q$ -version §17.6(ii)
- very-well-poised §17.4(iv)
- well-poised §17.4(iv)
- Zeilberger–Bressoud theorem §17.14
$q$ -hypergeometric orthogonal polynomials §18.27(i)
$q$ -integrals §17.2(v)
$q$ -Laguerre polynomials §18.27(v)
- applications §17.17
- asymptotic approximations to zeros §18.29

$q$ -Leibniz rule §17.2(iv)
$q$ -multinomial coefficient §26.16
$q$ -Pochhammer symbol §18.1(i)
$q$ -product §18.1(i)
$q$ -Racah polynomials §18.28(viii), see also Askey–Wilson class orthogonal polynomials.
- applications
  - coding theory §18.38(iii)
- relation to $q$ -hypergeometric function §18.28(viii)
$q$ -series
- classification §17.4(iv)
$q$ -sine function §17.3(i)
$q$ -Stirling numbers §17.3(iii)
$q^{-1}$ -Al-Salam–Chihara polynomials §18.28(iv)
quadratic characters
- number theory §27.9
quadratic equations §1.11(iii)
quadratic reciprocity law
- number theory §27.9
quadratic transformations
- classical orthogonal polynomials §18.7(ii)
- general orthogonal polynomials §18.2(vii)
quadrature §3.5—§3.5(ix)
- contour integrals §3.5(ix), §3.5(viii)—§3.5(viii)
- interpolatory rules (or formulas), see also Gauss quadrature.
  - Clenshaw–Curtis §3.5(iv)
  - closed §3.5(iv)
  - error term §3.5(iv)
  - Fejér’s §3.5(iv)
  - midpoint §3.5(iv)
  - Newton–Cotes §3.5(iv)
  - nodes §3.5(iv)
  - open §3.5(iv)
  - weight function §3.5(iv)
- oscillatory integrals
  - Clenshaw–Curtis formula (extended) §3.5(vii)
  - Filon’s rule §3.5(vii)
  - Longman’s method §3.5(vii)
  - multidimensional §3.5(vii)
- Romberg integration §3.5(iii)
- Simpson’s rule
  - composite §3.5(ii), §3.5(iv)
  - elementary §3.5(ii), §3.5(iv)
- steepest-descent paths §3.5(ix)—§3.5(ix)
- trapezoidal rule
  - composite §3.5(i), §3.5(iv), Table 3.5.20
  - elementary §3.5(i), §3.5(iv)
  - improved §3.5(i)
- via classical orthogonal polynomials §18.38(i)
quantum chemistry
- generalized exponential integral §8.24(iii)
- incomplete gamma functions §8.24(i)
quantum chromo-dynamics
- hypergeometric function §15.18
quantum field theory
- modular functions §23.21
- Riemann zeta function §25.17
quantum gravity
- Painlevé transcendents §32.16
quantum groups
- $q$ -series §17.17
quantum mechanics §18.39(i)
- associated Legendre functions §14.31(iii)
- canonical integrals §36.14(iii)
- Hermite EOP’s §18.39(i)
- Hermite polynomials §18.39(i)
- Heun functions §31.17(ii)
- Mathieu functions 3rd item, 6th item
- nonrelativistic
  - gamma function §5.20
- one-dimensional systems §18.39(i), §18.39(i)
- parabolic cylinder functions §12.17
- Romanovski–Bessel polynomials §18.39(i)
- Struve functions §11.12
- three-dimensional systems §18.39(ii)
- Whittaker functions §13.28(i)
quantum probability distributions
- Euler polynomials §24.18
quantum scattering
- hypergeometric function §15.18
quantum spin models
- Painlevé transcendents §32.16
quantum spins
- Heun’s equation §31.17(i)
quantum stationary states §18.39(i)
quantum superposition principle §18.39(i)
quantum systems
- Heun’s equation §31.17(ii)
quantum wave-packets
- theta functions §20.13
quark-gluon plasma
- Bernoulli polynomials §24.18
quartic equations §1.11(iii)
quartic oscillator
- Jacobian elliptic functions §22.19(ii)
quasiconformal mapping
- complete elliptic integrals §15.17(ii)
- hypergeometric function §15.17(ii)
queueing theory
- incomplete gamma functions §8.23
quintic equations
- modular functions §23.20(iv)
quotient-difference algorithm §3.10(ii)
- rhombus rule §3.10(ii)
- stability §3.10(ii)
quotient-difference scheme §3.10(ii)

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