Kelvin-function analogs
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11: 28.30 Expansions in Series of Eigenfunctions
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βΊFor analogous results to those of §28.19, see Schäfke (1960, 1961b), and Meixner et al. (1980, §1.1.11).
12: Bille C. Carlson
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βΊIn Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions.
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13: 6.4 Analytic Continuation
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6.4.4
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14: 6.2 Definitions and Interrelations
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Hyperbolic Analogs of the Sine and Cosine Integrals
…15: 24.16 Generalizations
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§24.16(i) Higher-Order Analogs
… βΊ§24.16(ii) Character Analogs
… βΊIn no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); -adic integer order Bernoulli numbers (Adelberg (1996)); -adic -Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).16: 17.1 Special Notation
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βΊThe main functions treated in this chapter are the basic hypergeometric (or -hypergeometric) function
, the bilateral basic hypergeometric (or bilateral -hypergeometric) function
, and the -analogs of the Appell functions
, , , and .
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17: 31.12 Confluent Forms of Heun’s Equation
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βΊThis is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i).
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18: 2.9 Difference Equations
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βΊThis situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)).
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βΊFor analogous results for difference equations of the form
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βΊFor asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999).
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19: 10.44 Sums
20: 28.7 Analytic Continuation of Eigenvalues
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βΊAnalogous statements hold for , , and , also for .
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