10 Bessel FunctionsModified Bessel Functions10.43 Integrals10.45 Functions of Imaginary Order

§10.44 Sums

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Permalink:

http://dlmf.nist.gov/10.44

See also:

Annotations for Ch.10

Contents

1. §10.44(i) Multiplication Theorem
2. §10.44(ii) Addition Theorems
3. §10.44(iii) Neumann-Type Expansions
4. §10.44(iv) Compendia

§10.44(i) Multiplication Theorem

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Keywords:

modified Bessel functions, multiplication theorem, sums

Notes:

For (10.44.1) combine (10.23.1) with (10.27.6) or with (10.27.8). Equations (10.44.2) are special cases of (10.23.1) and (10.44.1) with $\lambda =\mathrm{i}$.

Permalink:

http://dlmf.nist.gov/10.44.i

See also:

Annotations for §10.44 and Ch.10

10.44.1		$${𝒵}_{\nu }\left(\lambda z\right)={\lambda }^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{({\lambda }^2-1)}^k{(\frac{1}{2}z)}^k}{k!}{𝒵}_{\nu \pm k}\left(z\right),$$
.
ⓘ Symbols: $!$: factorial (as in $n!$), ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.51 Referenced by: §10.44(i), §10.66 Permalink: http://dlmf.nist.gov/10.44.E1 Encodings: TeX, pMML, png See also: Annotations for §10.44(i), §10.44 and Ch.10

If $𝒵=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary.

Examples

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See also:

Annotations for §10.44(i), §10.44 and Ch.10

10.44.2		$I_{\nu }\left(z\right)$	$={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{z^k}{k!}}{J}_{\nu +k}\left(z\right),$
		$J_{\nu }\left(z\right)$	$={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{z^k}{k!}}{I}_{\nu +k}\left(z\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.52 Referenced by: §10.44(i) Permalink: http://dlmf.nist.gov/10.44.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.44(i), §10.44(i), §10.44 and Ch.10

§10.44(ii) Addition Theorems

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Keywords:

addition theorems, modified Bessel functions, sums

Notes:

Combine (10.23.2) and (10.27.1) with (10.27.6) or with (10.27.8).

Referenced by:

§28.27

Permalink:

http://dlmf.nist.gov/10.44.ii

See also:

Annotations for §10.44 and Ch.10

Neumann’s Addition Theorem

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Keywords:

Neumann’s addition theorem, modified Bessel functions

See also:

Annotations for §10.44(ii), §10.44 and Ch.10

10.44.3		$${𝒵}_{\nu }\left(u\pm v\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{(\pm 1)}^k{𝒵}_{\nu +k}\left(u\right)I_k\left(v\right),$$
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ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.44.E3 Encodings: TeX, pMML, png See also: Annotations for §10.44(ii), §10.44(ii), §10.44 and Ch.10

The restriction is unnecessary when $𝒵=I$ and $\nu$ is an integer.

Graf’s and Gegenbauer’s Addition Theorems

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Keywords:

Gegenbauer’s addition theorem, Graf’s addition theorem, modified Bessel functions

See also:

Annotations for §10.44(ii), §10.44 and Ch.10

For results analogous to (10.23.7) and (10.23.8) see Watson (1944, §§11.3 and 11.41).

§10.44(iii) Neumann-Type Expansions

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Keywords:

Neumann-type expansions, expansions in series of, modified Bessel functions, sums

Notes:

Combine (10.23.15)–(10.23.17) with (10.27.6), (10.27.8), and (10.4.3).

Permalink:

http://dlmf.nist.gov/10.44.iii

See also:

Annotations for §10.44 and Ch.10

10.44.4		$${\left(\frac{1}{2}z\right)}^{\nu }=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{(\nu +2k)\mathrm{\Gamma }\left(\nu +k\right)}{k!}{I}_{\nu +2k}\left(z\right),$$
$\nu \ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.44.E4 Encodings: TeX, pMML, png See also: Annotations for §10.44(iii), §10.44 and Ch.10

10.44.5		$$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+2\sum _{k=1}^{\mathrm{\infty }}\frac{I_{2k}\left(z\right)}{k},$$
ⓘ Symbols: $\gamma$: Euler’s constant, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable A&S Ref: 9.6.54 Permalink: http://dlmf.nist.gov/10.44.E5 Encodings: TeX, pMML, png See also: Annotations for §10.44(iii), §10.44 and Ch.10

10.44.6		$$K_n\left(z\right)=\begin{array}{l}\frac{n!{(\frac{1}{2}z)}^{-n}}{2}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)}\\ +{(-1)}^{n-1}\left(\ln \left(\frac{1}{2}z\right)-\psi \left(n+1\right)\right)I_n\left(z\right)+{(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{(n+2k)I_{n+2k}\left(z\right)}{k(n+k)},\end{array}$$
ⓘ Symbols: $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 9.6.53 Permalink: http://dlmf.nist.gov/10.44.E6 Encodings: TeX, pMML, png See also: Annotations for §10.44(iii), §10.44 and Ch.10

where $\gamma$ is Euler’s constant and $\psi ={\mathrm{\Gamma }}^{\prime }/\mathrm{\Gamma }$ (§5.2).

§10.44(iv) Compendia

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Keywords:

compendia, integrals of modified Bessel functions, modified Bessel functions, sums

Permalink:

http://dlmf.nist.gov/10.44.iv

See also:

Annotations for §10.44 and Ch.10

For collections of sums and series involving modified Bessel functions see Erdélyi et al. (1953b, §7.15), Hansen (1975), and Prudnikov et al. (1986b, pp. 691–700).

10.43 Integrals10.45 Functions of Imaginary Order

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