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10 Bessel FunctionsSpherical Bessel Functions

§10.60 Sums

Contents
  1. §10.60(i) Addition Theorems
  2. §10.60(ii) Duplication Formulas
  3. §10.60(iii) Other Series
  4. §10.60(iv) Compendia

§10.60(i) Addition Theorems

Define u, v, w, and α as in §10.23(ii). Then with Pn again denoting the Legendre polynomial of degree n,

10.60.1 cosww=n=0(2n+1)𝗃n(v)𝗒n(u)Pn(cosα),
|ve±iα|<|u|.
10.60.2 sinww=n=0(2n+1)𝗃n(v)𝗃n(u)Pn(cosα).
10.60.3 eww=2πn=0(2n+1)𝗂n(1)(v)𝗄n(u)Pn(cosα),
|ve±iα|<|u|.

§10.60(ii) Duplication Formulas

10.60.4 𝗃n(2z)=n!zn+1k=0n2n2k+1k!(2nk+1)!𝗃nk(z)𝗒nk(z),
10.60.5 𝗒n(2z)=n!zn+1k=0nnk+12k!(2nk+1)!(𝗃nk2(z)𝗒nk2(z)),
10.60.6 𝗄n(2z)=1πn!zn+1k=0n(1)k2n2k+1k!(2nk+1)!𝗄nk2(z).

§10.60(iii) Other Series

10.60.7 eizcosα =n=0(2n+1)in𝗃n(z)Pn(cosα),
10.60.8 ezcosα =n=0(2n+1)𝗂n(1)(z)Pn(cosα),
10.60.9 ezcosα=n=0(1)n(2n+1)𝗂n(1)(z)Pn(cosα).
10.60.10 J0(zsinα)=n=0(4n+1)(2n)!22n(n!)2𝗃2n(z)P2n(cosα).
10.60.11 n=0𝗃n2(z)=Si(2z)2z.

For Si see §6.2(ii).

10.60.12 n=0(2n+1)𝗃n2(z) =1,
10.60.13 n=0(1)n(2n+1)𝗃n2(z) =sin(2z)2z,
10.60.14 n=0(2n+1)(𝗃n(z))2 =13.

For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000).

§10.60(iv) Compendia

For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). See also Watson (1944, Chapters 11 and 16).