# §10.23 Sums

## §10.23(i) Multiplication Theorem

 10.23.1 $\mathscr{C}_{\nu}\left(\lambda z\right)=\lambda^{\pm\nu}\sum_{k=0}^{\infty}% \frac{(\mp 1)^{k}(\lambda^{2}-1)^{k}(\tfrac{1}{2}z)^{k}}{k!}\mathscr{C}_{\nu% \pm k}\left(z\right),$ $|\lambda^{2}-1|<1$. ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.74 Referenced by: §10.44(i), §10.66 Permalink: http://dlmf.nist.gov/10.23.E1 Encodings: TeX, pMML, png See also: Annotations for §10.23(i), §10.23 and Ch.10

If $\mathscr{C}=J$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary.

 10.23.2 $\mathscr{C}_{\nu}\left(u\pm v\right)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu% \mp k}\left(u\right)J_{k}\left(v\right),$ $|v|<|u|$. ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $k$: nonnegative integer and $\nu$: complex parameter A&S Ref: 9.1.75 Referenced by: §10.23(iii), §10.23(ii), §10.44(ii), §10.66 Permalink: http://dlmf.nist.gov/10.23.E2 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

The restriction $|v|<|u|$ is unnecessary when $\mathscr{C}=J$ and $\nu$ is an integer. Special cases are:

 10.23.3 ${J_{0}}^{2}\left(z\right)+2\sum_{k=1}^{\infty}{J_{k}}^{2}\left(z\right)=1,$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $k$: nonnegative integer and $z$: complex variable A&S Ref: 9.1.76 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.23.E3 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10
 10.23.4 $\sum_{k=0}^{2n}(-1)^{k}J_{k}\left(z\right)J_{2n-k}\left(z\right)\\ +2\sum_{k=1}^{\infty}J_{k}\left(z\right)J_{2n+k}\left(z\right)=0,$ $n\geq 1$, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 9.1.77 Permalink: http://dlmf.nist.gov/10.23.E4 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10
 10.23.5 $\sum_{k=0}^{n}J_{k}\left(z\right)J_{n-k}\left(z\right)+2\sum_{k=1}^{\infty}(-1% )^{k}J_{k}\left(z\right)J_{n+k}\left(z\right)=J_{n}\left(2z\right).$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 9.1.78 Permalink: http://dlmf.nist.gov/10.23.E5 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

### Graf’s and Gegenbauer’s Addition Theorems

Define

 10.23.6 $\displaystyle w$ $\displaystyle=\sqrt{u^{2}+v^{2}-2uv\cos\alpha},$ $\displaystyle u-v\cos\alpha$ $\displaystyle=w\cos\chi,$ $\displaystyle v\sin\alpha$ $\displaystyle=w\sin\chi,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function and $\sin\NVar{z}$: sine function Permalink: http://dlmf.nist.gov/10.23.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

the branches being continuous and chosen so that $w\to u$ and $\chi\to 0$ as $v\to 0$. If $u$, $v$ are real and positive and $0\leq\alpha\leq\pi$, then $w$ and $\chi$ are real and nonnegative, and the geometrical relationship is shown in Figure 10.23.1.

 10.23.7 $\mathscr{C}_{\nu}\left(w\right)\selection{\cos\\ \sin}(\nu\chi)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu+k}\left(u\right)J_{k}% \left(v\right)\selection{\cos\\ \sin}(k\alpha),$ $|ve^{\pm i\alpha}|<|u|$.
 10.23.8 $\frac{\mathscr{C}_{\nu}\left(w\right)}{w^{\nu}}=2^{\nu}\Gamma\left(\nu\right)% \*\sum_{k=0}^{\infty}(\nu+k)\frac{\mathscr{C}_{\nu+k}\left(u\right)}{u^{\nu}}% \frac{J_{\nu+k}\left(v\right)}{v^{\nu}}C^{(\nu)}_{k}\left(\cos\alpha\right),$ $\nu\neq 0,-1,\dots$, $|ve^{\pm i\alpha}|<|u|$,

where $C^{(\nu)}_{k}\left(\cos\alpha\right)$ is Gegenbauer’s polynomial (§18.3). The restriction $|ve^{\pm i\alpha}|<|u|$ is unnecessary in (10.23.7) when $\mathscr{C}=J$ and $\nu$ is an integer, and in (10.23.8) when $\mathscr{C}=J$.

The degenerate form of (10.23.8) when $u=\infty$ is given by

 10.23.9 $e^{iv\cos\alpha}=\frac{\Gamma\left(\nu\right)}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k% =0}^{\infty}(\nu+k)i^{k}J_{\nu+k}\left(v\right)C^{(\nu)}_{k}\left(\cos\alpha% \right),$ $\nu\neq 0,-1,\dots$.

### Partial Fractions

For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005).

## §10.23(iii) Series Expansions of Arbitrary Functions

### Neumann’s Expansion

 10.23.10 $f(z)=a_{0}J_{0}\left(z\right)+2\sum_{k=1}^{\infty}a_{k}J_{k}\left(z\right),$ $|z|, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable, $a_{k}$ and $f(t)$ A&S Ref: 9.1.82 Referenced by: §10.23(iii) Permalink: http://dlmf.nist.gov/10.23.E10 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

where $c$ is the distance of the nearest singularity of the analytic function $f(z)$ from $z=0$,

 10.23.11 $a_{k}=\frac{1}{2\pi i}\int_{|t|=c^{\prime}}f(t)O_{k}\left(t\right)\,\mathrm{d}t,$ $0, ⓘ Defines: $a_{k}$ (locally) Symbols: $O_{\NVar{n}}\left(\NVar{x}\right)$: Neumann’s polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: nonnegative integer, $z$: complex variable and $f(t)$ A&S Ref: 9.1.83 Referenced by: Erratum (V1.1.2) for Equation (10.23.11) Permalink: http://dlmf.nist.gov/10.23.E11 Encodings: TeX, pMML, png Errata (effective with 1.1.2): Originally the contour of integration written incorrectly as $|z|=c^{\prime}$, has been corrected to be $|t|=c^{\prime}$. Reported 2021-03-22 by Mark Dunster See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

and $O_{k}\left(t\right)$ is Neumann’s polynomial, defined by the generating function:

 10.23.12 $\frac{1}{t-z}=J_{0}\left(z\right)O_{0}\left(t\right)+2\sum_{k=1}^{\infty}J_{k}% \left(z\right)O_{k}\left(t\right),$ $|z|<|t|$. ⓘ Defines: $O_{\NVar{n}}\left(\NVar{x}\right)$: Neumann’s polynomial Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $n$: integer, $k$: nonnegative integer, $x$: real variable and $z$: complex variable A&S Ref: 9.1.84 Permalink: http://dlmf.nist.gov/10.23.E12 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

$O_{n}\left(t\right)$ is a polynomial of degree $n+1$ in $\ifrac{1}{t}:O_{0}\left(t\right)=1/t$ and

 10.23.13 $O_{n}\left(t\right)=\frac{1}{4}\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\frac% {(n-k-1)!n}{k!}\left(\frac{2}{t}\right)^{n-2k+1},$ $n=1,2,\dotsc$.

For the more general form of expansion

 10.23.14 $z^{\nu}f(z)=a_{0}J_{\nu}\left(z\right)+2\sum_{k=1}^{\infty}a_{k}J_{\nu+k}\left% (z\right)$

see Watson (1944, §16.13), and for further generalizations see Watson (1944, Chapter 16) and Erdélyi et al. (1953b, §7.10.1).

### Examples

 10.23.15 $(\tfrac{1}{2}z)^{\nu}=\sum_{k=0}^{\infty}\frac{(\nu+2k)\Gamma\left(\nu+k\right% )}{k!}J_{\nu+2k}\left(z\right),$ $\nu\neq 0,-1,-2,\dots$,
 10.23.16 $Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)+\gamma% \right)J_{0}\left(z\right)-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{J_{2k% }\left(z\right)}{k},$
 10.23.17 $Y_{n}\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(% \tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}+\frac{2}{\pi}\left(\ln\left(% \tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)J_{n}\left(z\right)-\frac{2}{% \pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)J_{n+2k}\left(z\right)}{k(n+k)},$

where $\gamma$ is Euler’s constant and $\psi\left(n+1\right)=\Gamma'\left(n+1\right)/\Gamma\left(n+1\right)$5.2).

Other examples are provided by (10.12.1)–(10.12.6), (10.23.2), and (10.23.7).

### Fourier–Bessel Expansion

Assume $f(t)$ satisfies

 10.23.18 $\int_{0}^{1}t^{\frac{1}{2}}|f(t)|\,\mathrm{d}t<\infty,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $f(t)$ Permalink: http://dlmf.nist.gov/10.23.E18 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

and define

 10.23.19 $a_{m}=\frac{2}{(J_{\nu+1}\left(j_{\nu,m}\right))^{2}}\int_{0}^{1}tf(t)J_{\nu}% \left(j_{\nu,m}t\right)\,\mathrm{d}t,$ $\nu\geq-\tfrac{1}{2}$,

where $j_{\nu,m}$ is as in §10.21(i). If $0, then

 10.23.20 $\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)=\sum_{m=1}^{\infty}a_{m}J_{\nu}\left(j_{% \nu,m}x\right),$

provided that $f(t)$ is of bounded variation (§1.4(v)) on an interval $[a,b]$ with $0. This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near $x=0$ and $x=1$.

As an example,

 10.23.21 $x^{\nu}=\sum_{m=1}^{\infty}\frac{2J_{\nu}\left(j_{\nu,m}x\right)}{j_{\nu,m}J_{% \nu+1}\left(j_{\nu,m}\right)},$ $\nu>0,0\leq x<1$.

(Note that when $x=1$ the left-hand side is 1 and the right-hand side is 0.)

### Other Series Expansions

For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). See also Schäfke (1960, 1961b).

## §10.23(iv) Compendia

For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).