10 Bessel Functions Spherical Bessel Functions 10.59 Integrals 10.61 Definitions and Basic Properties

§10.60 Sums

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Keywords:: spherical Bessel functions, sums
Referenced by:: §10.47(i), Erratum (V1.1.6) for Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols
Permalink:: http://dlmf.nist.gov/10.60
See also:: Annotations for Ch.10

§10.60(i) Addition Theorems
§10.60(ii) Duplication Formulas
§10.60(iii) Other Series
§10.60(iv) Compendia

§10.60(i) Addition Theorems

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Keywords:: addition theorems, spherical Bessel functions, sums
Notes:: For (10.60.1)–(10.60.3) use (10.23.8) with $\nu=\tfrac{1}{2}$ and $\mathscr{C}=Y,J,{H^{(1)}}$ ; subsequently apply (10.47.12) and (10.47.13) in the case of (10.60.3).
Permalink:: http://dlmf.nist.gov/10.60.i
See also:: Annotations for §10.60 and Ch.10

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		$\frac{\cos w}{w}=-\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)\mathsf% {y}_{n}\left(u\right)P_{n}\left(\cos\alpha\right),$
$\|ve^{\pm i\alpha}\|<\|u\|$ .
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : 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		$\frac{\sin w}{w}=\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)\mathsf{% j}_{n}\left(u\right)P_{n}\left(\cos\alpha\right).$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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		$\frac{e^{-w}}{w}=\frac{2}{\pi}\sum_{n=0}^{\infty}(2n+1){\mathsf{i}^{(1)}_{n}}% \left(v\right)\mathsf{k}_{n}\left(u\right)P_{n}\left(\cos\alpha\right),$
$\|ve^{\pm i\alpha}\|<\|u\|$ .
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : 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(ii) Duplication Formulas

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Keywords:: duplication formulas, spherical Bessel functions, sums
Notes:: For (10.60.4) set $\mathscr{C}_{\nu}=Y_{\nu}$ , $u=v=z$ , $\nu=-n-\frac{1}{2}$ , and $\alpha=\pi$ in (10.23.8). Then refer to (10.47.3), (10.47.4), and also apply the following results obtained from Table 18.6.1: $C^{(-n-\frac{1}{2})}_{k}\left(-1\right)$ equals $\ifrac{(2n+1)!}{(k!(2n+1-k)!)}$ when $k=0,1,\dots,2n+1$ , and equals $0$ when $k=2n+2,2n+3,\dots$ . For (10.60.5) use the same procedure, but with $\mathscr{C}_{\nu}=J_{\nu}$ . Lastly, (10.60.6) follows by combining (10.60.4) and (10.60.5) with §10.47(iv).
Permalink:: http://dlmf.nist.gov/10.60.ii
See also:: Annotations for §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 $n!$ ), $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : 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 $n!$ ), $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : 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 $n!$ ), $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : 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(iii) Other Series

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Notes:: For (10.60.7)–(10.60.9) see Watson (1944, pp.368–369). For (10.60.10) use Watson (1944, p. 370, Eq. (9)) with $\nu=\tfrac{1}{2}$ , $\phi=\alpha$ , $\phi^{\prime}=\tfrac{1}{2}\pi$ ; also Eq. (18.7.9). For (10.60.11) see Watson (1944, p. 152). For (10.60.12) and (10.60.13) substitute $u=v=z$ , with $\alpha=0$ and $\pi$ , into (10.60.2). For (10.60.14) see Vavreck and Thompson (1984).
Permalink:: http://dlmf.nist.gov/10.60.iii
See also:: Annotations for §10.60 and Ch.10

10.60.7	$\displaystyle e^{iz\cos\alpha}$	$\displaystyle=\sum_{n=0}^{\infty}(2n+1)i^{n}\mathsf{j}_{n}\left(z\right)P_{n}% \left(\cos\alpha\right),$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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	$\displaystyle e^{z\cos\alpha}$	$\displaystyle=\sum_{n=0}^{\infty}(2n+1){\mathsf{i}^{(1)}_{n}}\left(z\right)P_{% n}\left(\cos\alpha\right),$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : 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		$e^{-z\cos\alpha}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1){\mathsf{i}^{(1)}_{n}}\left(% z\right)P_{n}\left(\cos\alpha\right).$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : 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		$J_{0}\left(z\sin\alpha\right)=\sum_{n=0}^{\infty}(4n+1)\frac{(2n)!}{2^{2n}(n!)% ^{2}}\mathsf{j}_{2n}\left(z\right)P_{2n}\left(\cos\alpha\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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		$\sum_{n=0}^{\infty}{\mathsf{j}_{n}}^{2}\left(z\right)=\frac{\operatorname{Si}% \left(2z\right)}{2z}.$
ⓘ Symbols: $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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 $\operatorname{Si}$ see §6.2(ii).

10.60.12	$\displaystyle\sum_{n=0}^{\infty}(2n+1){\mathsf{j}_{n}}^{2}\left(z\right)$	$\displaystyle=1,$
ⓘ Symbols: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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	$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}(2n+1){\mathsf{j}_{n}}^{2}\left(z\right)$	$\displaystyle=\frac{\sin\left(2z\right)}{2z},$
ⓘ Symbols: $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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	$\displaystyle\sum_{n=0}^{\infty}(2n+1)(\mathsf{j}_{n}'\left(z\right))^{2}$	$\displaystyle=\tfrac{1}{3}.$
ⓘ Symbols: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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).

§10.60(iv) Compendia

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Keywords:: compendia, spherical Bessel functions, sums
Permalink:: http://dlmf.nist.gov/10.60.iv
See also:: Annotations for §10.60 and Ch.10

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 (2015, §§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).

10.59 Integrals 10.61 Definitions and Basic Properties

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§10.60 Sums

Contents

§10.60(i) Addition Theorems

§10.60(ii) Duplication Formulas

§10.60(iii) Other Series

§10.60(iv) Compendia