§10.44 Sums

10.44.1		$${𝒵}_{\nu }\left(\lambda z\right)={\lambda }^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{({\lambda }^2-1)}^k{(\frac{1}{2}z)}^k}{k!}{𝒵}_{\nu \pm k}\left(z\right),$$
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ⓘ 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.

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).

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).