Jacobi–Abel addition theorem
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11: 1.15 Summability Methods
Abel Summability
… ►Abel Means
… ► is a harmonic function in polar coordinates (1.9.27), and … ►Here is the Abel (or Poisson) sum of , and has the series representation … ►Abel Summability
…12: Errata
Equation (16.16.5_5).
Originally the term was given incorrectly as in this equation and in the line above. Additionally, for improved clarity, the modulus has been defined in the line above.
Reported 2014-05-02 by Svante Janson.
Originally a minus sign was missing in the entries for and in the second column (headed ). The correct entries are and . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.
Reported 2014-02-28 by Svante Janson.
Originally the term was given incorrectly as .
Reported 2014-02-28 by Svante Janson.
13: 18.10 Integral Representations
Jacobi
… ►for the Jacobi, Laguerre, and Hermite polynomials. … ► …14: 22.6 Elementary Identities
§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
► …15: 18.30 Associated OP’s
§18.30(i) Associated Jacobi Polynomials
… ►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). For other results for associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). … ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real -axis each multiplied by the polynomial product evaluated at the corresponding values of , as in (18.2.3). … ►Markov’s Theorem
…16: 18.1 Notation
Classical OP’s
►Jacobi: .
Big -Jacobi: .
Little -Jacobi: .