Digital Library of Mathematical Functions
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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.24 Series


§13.24(i) Expansions in Series of Whittaker Functions

For expansions of arbitrary functions in series of Mκ,μ(z) functions see Schäfke (1961b).

§13.24(ii) Expansions in Series of Bessel Functions

For z, and again with the notation of §§10.2(ii) and 10.25(ii),

13.24.1 Mκ,μ(z)=Γ(κ+μ)22κ+2μz12-κs=0(-1)s(2κ+2μ)s(2κ)s(1+2μ)ss!(κ+μ+s)Iκ+μ+s(12z),


13.24.2 1Γ(1+2μ)Mκ,μ(z)=22μzμ+12s=0ps(μ)(z)(2κz)-2μ-sJ2μ+s(2κz),

where p0(μ)(z)=1, p1(μ)(z)=16z2, and higher polynomials ps(μ)(z) are defined by

13.24.3 exp(-12z(cotht-1t))(tsinht)1-2μ=s=0ps(μ)(z)(-tz)s.

(13.18.8) is a special case of (13.24.1).

Additional expansions in terms of Bessel functions are given in Luke (1959). See also López (1999).

For other series expansions see Prudnikov et al. (1990, §6.6). See also §13.26.