Jacobi fraction (J-fraction)
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1: 22.16 Related Functions
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§22.16(i) Jacobi’s Amplitude () Function
… ►§22.16(ii) Jacobi’s Epsilon Function
►Integral Representations
… ►§22.16(iii) Jacobi’s Zeta Function
… ►Properties
…2: 18.3 Definitions
§18.3 Definitions
►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of for Jacobi polynomials, in powers of for the other cases). … ►Jacobi on Other Intervals
…3: 3.10 Continued Fractions
§3.10 Continued Fractions
… ►Stieltjes Fractions
… ►Jacobi Fractions
… ►is called a Jacobi fraction (-fraction). …For the same function , the convergent of the Jacobi fraction (3.10.11) equals the convergent of the Stieltjes fraction (3.10.6). …4: 20.11 Generalizations and Analogs
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►In the case identities for theta functions become identities in the complex variable , with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
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►This is Jacobi’s inversion problem of §20.9(ii).
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►Each provides an extension of Jacobi’s inversion problem.
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►For , , and , define twelve combined theta functions
by
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5: 22.8 Addition Theorems
6: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.2
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22.12.8
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22.12.11
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22.12.13
7: 12.6 Continued Fraction
§12.6 Continued Fraction
►For a continued-fraction expansion of the ratio see Cuyt et al. (2008, pp. 340–341).8: 22.6 Elementary Identities
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22.6.2
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22.6.5
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22.6.8
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