Index B

Bäcklund transformations §18.38(ii)
- Painlevé transcendents §32.7—§32.7(viii)
backward recursion §3.6(ii)
Bailey’s ${{}_{2}F_{1}}(-1)$ sum
- $q$ -analog §17.7(i)
Bailey’s ${{}_{2}\psi_{2}}$ transformations
- bilateral $q$ -hypergeometric function §17.10
Bailey’s ${{}_{4}F_{3}}(1)$ sum
- $q$ -analogs (first and second) §17.7(iii)
Bailey’s bilateral summations
- bilateral $q$ -hypergeometric function §17.8
band spectra
- second order differential operators §1.18(vii)
bandlimited functions §30.15(iii)
Bannai–Ito polynomials §18.28(xi)
Barnes’ beta integral §5.13
Barnes’ $G$ -function
- asymptotic expansion §5.17
- definition §5.17
- infinite product §5.17
- integral representation §5.17
- recurrence relation §5.17
Barnes’ integral
- Ferrers functions §14.17(ii)
Bartky’s transformation
- Bulirsch’s elliptic integrals §19.2(iii)
- symmetric elliptic integrals §19.22(i)
Barycentric form of Lagrange interpolation §3.3(i)
basic elliptic integrals §19.29(ii)
basic hypergeometric functions, see bilateral $q$ -hypergeometric function and $q$ -hypergeometric function.
Basset’s integral
- modified Bessel functions §10.32(i)
Bateman-type sums §18.18(vi)
Bell numbers
- asymptotic approximations §26.7(iv)
- definition §26.7(i)
- generating function §26.7(ii)
- recurrence relation §26.7(iii)
- table Table 26.7.1
Bernoulli monosplines §24.17(ii)
Bernoulli numbers §24.1
- arithmetic properties §24.10
- asymptotic approximations §24.11
- computation §24.19(i)
- definition §24.2
- degenerate §24.16(i)
- explicit formulas §24.6—§24.6
- factors §24.10(iv)
- finite expansions §24.4(iv)
- generalizations §24.16, §24.16(iii)
- generating function §24.2
- identities §24.5(ii), §24.5(iii)
- inequalities §24.9
- integral representations §24.7(i)
- inversion formulas §24.5(iii)
- irregular pairs §24.19(ii)
- Kummer congruences §24.10(ii)
- notation §24.1
- of the second kind §24.16(i)
- recurrence relations
  - linear §24.5
  - quadratic and higher order §24.14(i), §24.14(ii)
- relations to
  - Eulerian numbers §24.4(ix)
  - Genocchi numbers §24.15(i)
  - Stirling numbers §24.15(iii)
  - tangent numbers §24.15(ii)
- sums §24.14
- tables Table 24.2.1, §24.20
Bernoulli polynomials §24.1
- applications
  - mathematical §24.17—§24.17(iii)
  - physical §24.18
- as Sheffer polynomials §18.2(xii)
- asymptotic approximations §24.11
- computation §24.19(i)
- definitions §24.2(i)
- derivative §24.4(vii)
- difference equation §24.4(i)
- explicit formulas §24.6
- finite expansions §24.4(iv)
- generalized §24.16, §24.16(iii)
- generating function §24.2(i)
- graphs Figure 24.3.1, Figure 24.3.1, Figure 24.3.1
- inequalities §24.9
- infinite series expansions
  - Fourier §24.8(i)
  - other §24.8(ii)
- integral representations §24.7(ii)
- integrals §24.13(i)
  - compendia §24.13(iii)
- Laplace transforms §24.13(iii)
- multiplication formulas §24.4(v)
- notation §24.1
- recurrence relations
  - linear §24.5
  - quadratic §24.14(i)
- relation to Eulerian numbers §24.4(ix)
- relation to Riemann zeta function §24.4(ix)
- representation as sums of powers §24.4(iii)
- special values §24.4(vi)
- sums §24.14
- symbolic operations §24.4(viii)
- symmetry §24.4(ii)
- tables Table 24.2.2
- zeros
  - complex §24.12(iii)
  - multiple §24.12(iv)
  - real §24.12(i)
Bernoulli’s lemniscate §19.30(iii)
Bernstein–Szegő polynomials §18.31
Bessel eigenfunction expansion
- Hankel transform §1.18(vi)
Bessel Functions
- Jacobi–Anger expansions §10.35
Bessel functions §10.1, see also cylinder functions, Hankel functions, Kelvin functions, modified Bessel functions, and spherical Bessel functions.
- addition theorems §10.23(ii)
- analytic continuation §10.11
- and eigenfunctions of Dunkl operator §18.38(iii)
- and zeros of Jacobi polynomials §18.16(ii)
- applications
  - asymptotic solutions of differential equations §10.72(i)—§10.72(iii)
  - electromagnetic scattering §10.73(i)
  - Helmholtz equation §10.73(i)—§10.73(i)
  - Laplace’s equation §10.73(i)—§10.73(i)
  - oscillation of chains §10.73(i)
  - oscillation of plates §10.73(i)
  - wave equation §10.73(i)
- approximations §10.76
- asymptotic expansions for large argument §10.17—§10.17(v)
  - error bounds §10.17(iii)—§10.17(v)
  - exponentially-improved §10.17(v)
- asymptotic expansions for large order §10.19—§10.20(iii)
  - asymptotic forms §10.19(i)
  - Debye’s expansions §10.19(ii)—§10.19(ii)
  - double asymptotic properties §10.20(iii), §10.41(v)
  - resurgence properties of coefficients §10.20(ii)
  - transition region §10.19(iii)
  - uniform §10.20—§10.20(iii)
- asymptotics of Jacobi polynomials §18.15(i)
- asymptotics of Laguerre polynomials §18.15(iv)
- branch conventions §10.2(ii)
- computation Ch.10—§10.74(v)
- computation by quadrature §3.5(viii)
- computation by recursion §3.6(vi)
- connection formulas §10.4
- contiguous §10.21(i)
- continued fractions §10.10
- cross-products §10.5, §10.6(iii)
  - zeros §10.21(x)
- definite integrals §9.11(iii)
- definitions §10.2—§10.2(ii)
- derivatives
  - asymptotic expansions for large argument §10.17(ii)
  - asymptotic expansions for large order §10.19(ii)—§10.19(iii)
  - explicit forms §10.6(ii)
  - uniform asymptotic expansions for large order §10.20(i)
  - with respect to order §10.15—§10.15
  - zeros, see zeros of Bessel functions.
- differential equations §10.13, §10.2(i), see also Bessel’s equation.
- Dirac delta §1.17(ii)
- envelope functions §2.8(iv)
- expansions in partial fractions §10.23(ii)
- expansions in series of §10.23(iii)—§10.23(iv)
- Fourier transform of ultraspherical polynomials §18.17(v)
- Fourier–Bessel expansion §10.23(iii)
- generalized §10.46
- generating functions §10.12
- graphics §10.3—§10.3(iii)
- incomplete §10.46
- inequalities §10.14
- infinite products §10.21(iii)
- integral representation of Laguerre polynomials §18.10(iv)
- integral representations
  - along the real line §10.9(i)—§10.9(i)
  - compendia §10.9(iv)
  - contour integrals §10.9(ii)—§10.9(ii)
  - Mellin–Barnes type §10.9(ii)
  - products §10.9(iii)—§10.9(iii)
- integrals, see also integrals of Bessel and Hankel functions and Hankel transform.
  - approximations §10.76(ii)
  - computation §10.74(vii)
  - tables §10.75(iv), §10.75(vii)
- limit of Jacobi polynomials §18.11(ii)
- limit of Laguerre polynomials §18.11(ii)
- limiting forms §10.7
- minimax rational approximation §3.11(iii)
- modulus and phase functions
  - asymptotic expansions for large argument §10.18(iii)—§10.18(iii)
  - basic properties §10.18(ii)
  - definitions §10.18(i)
  - graphics §10.3(i)—§10.3(i)
  - relation to zeros §10.21(ii)
- monotonicity §10.14
- multiplication theorem §10.23(i)
- notation §10.1
- of imaginary argument, see modified Bessel functions.
- of imaginary order
  - applications §10.24
  - definitions §10.24
  - graphs §10.3(iii)—§10.3(iii)
  - limiting forms §10.24—§10.24
  - numerically satisfactory pairs §10.24
  - uniform asymptotic expansions for large order §10.24
  - zeros §10.24
- of matrix argument §35.5
  - applications §35.9
  - asymptotic approximations §35.5(iii)
  - definitions §35.5(i)
  - notation §35.1
  - of the first and second kinds §35.1
  - properties §35.5(ii)
  - relations to confluent hypergeometric functions of matrix argument §35.6(iii)
- of the first kind §10.2(ii)
- of the first, second, and third kinds §10.2(ii)—§10.2(iii)
- of the second kind §10.2(ii)
- of the third kind §10.2(ii)
- orthogonality §10.22(ii), §10.22(iv)
- power series §10.8
- principal branches (or values) §10.2(ii)—§10.2(iii)
- recurrence relations §10.6(i)—§10.6(iii)
- relations to other functions
  - Airy functions §9.6(i)—§9.6(ii)
  - confluent hypergeometric functions §10.16
  - elementary functions §10.16
  - generalized Airy functions §9.13(i)
  - generalized hypergeometric functions §10.16
  - parabolic cylinder functions §10.16, §12.14(vii)
- sums §10.23(i)—§10.23(iv)
  - addition theorems §10.23(ii)—§10.23(ii)
  - compendia §10.23(iv)
  - expansions in series of Bessel functions §10.23(iii)—§10.23(iii)
  - multiplication theorem §10.23(i)
- tables §10.75—§10.75(iii)
- Wronskians §10.5
- zeros, see zeros of Bessel functions.

Bessel polynomials §10.49(ii), §18.34—§18.34(iii)
- asymptotic expansions §18.34(iii)
- definition §18.34(i)
- differential equations §18.34(iii)
- generalized §18.34(i)
- orthogonality properties §18.34(ii)
- recurrence relations §18.34(i)
- relations to other functions
  - complex orthogonal polynomials §3.5(viii)
  - confluent hypergeometric functions §18.34(i)
  - generalized hypergeometric functions §18.34(i)
  - Jacobi polynomials §18.34(iii)
  - Laguerre polynomials §18.34(i)
  - modified Bessel functions §18.34(i)
  - Whittaker functions §18.34(i)
Bessel transform, see Hankel transform.
Bessel’s equation §10.2(i)
- inhomogeneous forms §11.10(ii), §11.2(ii), §11.9(i)
- numerically satisfactory solutions §10.2(iii)
- singularities §10.2(i)
- standard solutions §10.2(ii)—§10.2(ii)
Bessel’s integral
- Bessel functions §10.9(i)
best uniform polynomial approximation §3.11(i)
best uniform rational approximation §3.11(iii)
beta distribution
- incomplete beta functions §8.23
beta function §5.12, see also incomplete beta functions.
- applications
  - physical §5.20—§5.20
- definition §5.12
- integral representations §5.12
  - multidimensional §5.14
- integrals §5.13
- multivariate, see multivariate beta function.
beta integrals §5.12—§5.13
Bickley function §10.43(iii)—§10.43(iv)
- applications §10.73(iv)
- approximations §10.76(iii)
biconfluent Heun equation §31.12
- application to Rossby waves §31.17(ii)
Bieberbach conjecture §16.23(iii), §18.38(ii)
- and Askey–Gasper inequality §18.38(ii)
bifurcation sets §36.4
- visualizations §36.4(ii)
big $q$ -Jacobi polynomials §18.27(iii)
biharmonic operator
- numerical approximation §3.4(iii)
bilateral basic hypergeometric function, see bilateral $q$ -hypergeometric function.
bilateral hypergeometric function §16.4(v)
bilateral $q$ -hypergeometric function
- Bailey’s ${{}_{2}\psi_{2}}$ transformations §17.10
- Bailey’s bilateral summations §17.8
- computation §17.18
- definition §17.4(ii)
- notation §17.1
- Ramanujan’s ${{}_{1}\psi_{1}}$ summation §17.8
- special cases §17.8—§17.8
- transformations §17.18
bilateral series §16.4(v)
bilinear transformation §1.9(iv)
- cross ratio §1.9(iv)
- SL $(2,\mathbb{Z})$ §23.15(i)
binary number system §3.1(i)
binary quadratic sieve
- number theory §27.18
Binet’s formula
- gamma function §5.9(i)
binomial coefficients
- definition §1.2(i)
- generating functions §26.3(ii)
- identities §26.3(iv)
- limiting form §26.3(v)
- recurrence relations §26.3(iii)
- relation to lattice paths §26.3(i)
- tables §26.21, Table 26.3.1, Table 26.3.2
binomial expansion §4.6(ii)
binomial theorem §1.2(i)
binomials §1.2(i)
black holes
- Heun functions §31.17(ii)
Bochner’s theorem item 1.
Bohr radius
- Coulomb functions §33.22(ii)
Bohr-Mollerup theorem
- gamma function §5.5(iv)
- $q$ -gamma function §5.18(ii)
Boole summation formula §24.17(i)
Borel summability §1.15(i)
Borel transform theory
- applications to asymptotic expansions §2.11(v)
Bose–Einstein condensates
- Lamé functions §29.19(i)
Bose–Einstein integrals
- computation §25.18(i)
- definition §25.12(iii)
- relation to polylogarithms §25.12(iii)
Bose–Einstein phase transition §25.17
bound-state problems
- hydrogenic §33.22(v)
- Whittaker functions §33.22(v)
boundary conditions
- second order linear differential operator §1.18(ix)
boundary conditions and the Weyl alternative
- mathematical background §1.18(ix)
boundary points §1.6(iv), §1.9(ii)
boundary-value methods or problems
- difference equations §3.6(iv), §3.6(vii)
- ordinary differential equations §3.7(iii)
- parabolic cylinder functions §12.17
bounded variation §1.4(v)
Boussinesq equation
- Painlevé transcendents §32.13(iii)
box
- plane partitions §26.12(i)
branch
- of multivalued function §1.10(vi), §4.2(i)
  - construction §1.10(vi)—§1.10(vi)
  - example §1.10(vi)
branch cut §4.2(i)
branch point §1.10(vi)
- movable §32.2(i)
Bromwich integral §3.5(viii)
Bulirsch’s elliptic integrals §19.2(iii)
- computation §19.36(ii)
- first, second, and third kinds §19.1
- notation §19.1
- relation to symmetric elliptic integrals §19.25(ii)

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