§10.60 Sums

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

10.60.1		$$\frac{\cos w}{w}=-\sum _{n=0}^{\mathrm{\infty }}(2n+1){\mathsf{j}}_n\left(v\right){\mathsf{y}}_n\left(u\right)P_n\left(\cos \alpha \right),$$
.
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind and $n$: integer A&S Ref: 10.1.46 (corrected) Referenced by: §10.60(i) Permalink: http://dlmf.nist.gov/10.60.E1 Encodings: TeX, pMML, png See also: Annotations for §10.60(i), §10.60 and Ch.10

10.60.2		$$\frac{\sin w}{w}=\sum _{n=0}^{\mathrm{\infty }}(2n+1){\mathsf{j}}_n\left(v\right){\mathsf{j}}_n\left(u\right)P_n\left(\cos \alpha \right).$$
ⓘ Symbols: $\cos z$: cosine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $\sin z$: sine function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer A&S Ref: 10.1.45 Referenced by: §10.60(iii) Permalink: http://dlmf.nist.gov/10.60.E2 Encodings: TeX, pMML, png See also: Annotations for §10.60(i), §10.60 and Ch.10

10.60.3		$$\frac{{\mathrm{e}}^{-w}}{w}=\frac{2}{\pi }\sum _{n=0}^{\mathrm{\infty }}(2n+1){\mathsf{i}}_n^{(1)}\left(v\right){\mathsf{k}}_n\left(u\right)P_n\left(\cos \alpha \right),$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${\mathsf{k}}_n\left(z\right)$: modified spherical Bessel function and $n$: integer A&S Ref: 10.2.35 (with condition added) Referenced by: §10.60(i) Permalink: http://dlmf.nist.gov/10.60.E3 Encodings: TeX, pMML, png See also: Annotations for §10.60(i), §10.60 and Ch.10

10.60.4		$${\mathsf{j}}_n\left(2z\right)=-n!z^{n+1}\sum _{k=0}^n\frac{2n-2k+1}{k!(2n-k+1)!}{\mathsf{j}}_{n-k}\left(z\right){\mathsf{y}}_{n-k}\left(z\right),$$
ⓘ Symbols: $!$: factorial (as in $n!$), ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 10.1.49 (corrected) Referenced by: §10.60(ii), Ch.10 Permalink: http://dlmf.nist.gov/10.60.E4 Encodings: TeX, pMML, png See also: Annotations for §10.60(ii), §10.60 and Ch.10

10.60.5		$${\mathsf{y}}_n\left(2z\right)=n!z^{n+1}\sum _{k=0}^n\frac{n-k+\frac{1}{2}}{k!(2n-k+1)!}\left({\mathsf{j}}_{n-k}^2\left(z\right)-{\mathsf{y}}_{n-k}^2\left(z\right)\right),$$
ⓘ Symbols: $!$: factorial (as in $n!$), ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable Referenced by: §10.60(ii) Permalink: http://dlmf.nist.gov/10.60.E5 Encodings: TeX, pMML, png See also: Annotations for §10.60(ii), §10.60 and Ch.10

10.60.6		$${\mathsf{k}}_n\left(2z\right)=\frac{1}{\pi }n!z^{n+1}\sum _{k=0}^n{(-1)}^k\frac{2n-2k+1}{k!(2n-k+1)!}{\mathsf{k}}_{n-k}^2\left(z\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathsf{k}}_n\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 10.2.38 Referenced by: §10.60(ii), Ch.10 Permalink: http://dlmf.nist.gov/10.60.E6 Encodings: TeX, pMML, png See also: Annotations for §10.60(ii), §10.60 and Ch.10

10.60.7		${\mathrm{e}}^{\mathrm{i}z\cos \alpha }$	$={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}(2n+1){\mathrm{i}}^n{\mathsf{j}}_n\left(z\right)P_n\left(\cos \alpha \right),$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.47 Referenced by: §10.60(iii), Ch.10 Permalink: http://dlmf.nist.gov/10.60.E7 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10
10.60.8		${\mathrm{e}}^{z\cos \alpha }$	$={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}(2n+1){\mathsf{i}}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right),$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable A&S Ref: 10.2.36 Permalink: http://dlmf.nist.gov/10.60.E8 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10

10.60.9		$${\mathrm{e}}^{-z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(2n+1){\mathsf{i}}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right).$$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable A&S Ref: 10.2.37 Referenced by: §10.60(iii), Ch.10 Permalink: http://dlmf.nist.gov/10.60.E9 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10

10.60.10		$$J_0\left(z\sin \alpha \right)=\sum _{n=0}^{\mathrm{\infty }}(4n+1)\frac{(2n)!}{2^{2n}{(n!)}^2}{\mathsf{j}}_{2n}\left(z\right)P_{2n}\left(\cos \alpha \right).$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\cos z$: cosine function, $!$: factorial (as in $n!$), $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $\sin z$: sine function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.48 Referenced by: §10.60(iii), Ch.10 Permalink: http://dlmf.nist.gov/10.60.E10 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10

10.60.11		$$\sum _{n=0}^{\mathrm{\infty }}{\mathsf{j}}_n^2\left(z\right)=\frac{\mathrm{Si}\left(2z\right)}{2z}.$$
ⓘ Symbols: $\mathrm{Si}\left(z\right)$: sine integral, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.52 Referenced by: §10.60(iii), Ch.10 Permalink: http://dlmf.nist.gov/10.60.E11 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10

For $\mathrm{Si}$ see §6.2(ii).

10.60.12		${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}(2n+1){\mathsf{j}}_n^2\left(z\right)$	$=1,$
ⓘ Symbols: ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.50 Referenced by: §10.60(iii) Permalink: http://dlmf.nist.gov/10.60.E12 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10
10.60.13		${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{(-1)}^n(2n+1){\mathsf{j}}_n^2\left(z\right)$	$={\displaystyle \frac{\sin \left(2z\right)}{2z}},$
ⓘ Symbols: $\sin z$: sine function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 10.1.51 Referenced by: §10.60(iii) Permalink: http://dlmf.nist.gov/10.60.E13 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10
10.60.14		${\displaystyle \sum _{n=0}^{\mathrm{\infty }}}(2n+1){({\mathsf{j}}_n^{\prime }\left(z\right))}^2$	$=\frac{1}{3}.$
ⓘ Symbols: ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable Referenced by: §10.60(iii) Permalink: http://dlmf.nist.gov/10.60.E14 Encodings: TeX, pMML, png See also: Annotations for §10.60(iii), §10.60 and Ch.10

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).

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).