# §10.23 Sums

## §10.23(i) Multiplication Theorem

 10.23.1 $\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(\lambda z\right)=\lambda^{\pm\nu}% \sum_{k=0}^{\infty}\frac{(\mp 1)^{k}(\lambda^{2}-1)^{k}(\tfrac{1}{2}z)^{k}}{k!% }\mathop{\mathscr{C}_{\nu\pm k}\/}\nolimits\!\left(z\right),$ $|\lambda^{2}-1|<1$. Symbols: $\mathop{\mathscr{C}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

## §10.23(ii) Addition Theorems

### Neumann’s Addition Theorem

 10.23.2 $\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(u\pm v\right)=\sum_{k=-\infty}^{% \infty}\mathop{\mathscr{C}_{\nu\mp k}\/}\nolimits\!\left(u\right)\mathop{J_{k}% \/}\nolimits\!\left(v\right),$ $|v|<|u|$. Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{\mathscr{C}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

 10.23.3 ${\mathop{J_{0}\/}\nolimits^{2}}\!\left(z\right)+2\sum_{k=1}^{\infty}{\mathop{J% _{k}\/}\nolimits^{2}}\!\left(z\right)=1,$ Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.23.4 $\sum_{k=0}^{2n}(-1)^{k}\mathop{J_{k}\/}\nolimits\!\left(z\right)\mathop{J_{2n-% k}\/}\nolimits\!\left(z\right)\\ +2\sum_{k=1}^{\infty}\mathop{J_{k}\/}\nolimits\!\left(z\right)\mathop{J_{2n+k}% \/}\nolimits\!\left(z\right)=0,$ $n\geq 1$, Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.23.5 $\sum_{k=0}^{n}\mathop{J_{k}\/}\nolimits\!\left(z\right)\mathop{J_{n-k}\/}% \nolimits\!\left(z\right)+2\sum_{k=1}^{\infty}(-1)^{k}\mathop{J_{k}\/}% \nolimits\!\left(z\right)\mathop{J_{n+k}\/}\nolimits\!\left(z\right)=\mathop{J% _{n}\/}\nolimits\!\left(2z\right).$ Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

Define

 10.23.6 $\displaystyle w$ $\displaystyle=\sqrt{u^{2}+v^{2}-2uv\mathop{\cos\/}\nolimits\alpha},$ $\displaystyle u-v\mathop{\cos\/}\nolimits\alpha$ $\displaystyle=w\mathop{\cos\/}\nolimits\chi,$ $\displaystyle v\mathop{\sin\/}\nolimits\alpha$ $\displaystyle=w\mathop{\sin\/}\nolimits\chi,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: 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)

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

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

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

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

### 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 $f(z)=a_{0}\mathop{J_{0}\/}\nolimits\!\left(z\right)+2\sum_{k=1}^{\infty}a_{k}% \mathop{J_{k}\/}\nolimits\!\left(z\right),$ $|z|, Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

 10.23.11 $a_{k}=\frac{1}{2\pi i}\int_{|z|=c^{\prime}}f(t)\mathop{O_{k}\/}\nolimits\!% \left(t\right)\mathrm{d}t,$ $0, Defines: $a_{k}$ (locally) Symbols: $\mathop{O_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Neumann’s polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $k$: nonnegative integer, $z$: complex variable and $f(t)$ A&S Ref: 9.1.83 Permalink: http://dlmf.nist.gov/10.23.E11 Encodings: TeX, pMML, png See also: Annotations for 10.23(iii)

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

 10.23.12 $\frac{1}{t-z}=\mathop{J_{0}\/}\nolimits\!\left(z\right)\mathop{O_{0}\/}% \nolimits\!\left(t\right)+2\sum_{k=1}^{\infty}\mathop{J_{k}\/}\nolimits\!\left% (z\right)\mathop{O_{k}\/}\nolimits\!\left(t\right),$ $|z|<|t|$. Defines: $\mathop{O_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Neumann’s polynomial Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

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

For the more general form of expansion

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

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

where $\gamma$ is Euler’s constant and $\mathop{\psi\/}\nolimits\!\left(n+1\right)=\mathop{\Gamma\/}\nolimits'\!\left(% n+1\right)/\mathop{\Gamma\/}\nolimits\!\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 of $x$, $\int$: integral and $f(t)$ Permalink: http://dlmf.nist.gov/10.23.E18 Encodings: TeX, pMML, png See also: Annotations for 10.23(iii)

and define

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

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

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

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

(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).