series representation
(0.003 seconds)
31—40 of 78 matching pages
31: 18.3 Definitions
§18.3 Definitions
… ►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). 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). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for are given in §18.5(iv). For explicit power series coefficients up to for these polynomials and for coefficients up to for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …32: 34.2 Definition: Symbol
33: 4.13 Lambert -Function
34: 34.4 Definition: Symbol
§34.4 Definition: Symbol
… ►where is defined as in §16.2. ►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).35: 8.27 Approximations
Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the -plane that exclude and are valid for .
Luke (1975, p. 103) gives Chebyshev-series expansions for and related functions for .
36: Bibliography D
37: 22.20 Methods of Computation
38: 6.18 Methods of Computation
39: 16.11 Asymptotic Expansions
§16.11(i) Formal Series
►For subsequent use we define two formal infinite series, and , as follows: …Explicit representations for the coefficients are given in Volkmer (2023). … ►The formal series (16.11.2) for converges if , and … ►Explicit representations for the coefficients are given in Volkmer and Wood (2014). …40: Errata
§4.13 has been enlarged. The Lambert -function is multi-valued and we use the notation , , for the branches. The original two solutions are identified via and .
Other changes are the introduction of the Wright -function and tree -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert -functions in the end of the section.