§10.44 Sums

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

§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
§10.44(iii) Neumann-Type Expansions
§10.44(iv) Compendia

§10.44(i) Multiplication Theorem

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 $\lambda=i$ .
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.44.i

10.44.1 $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(\lambda z\right)=\lambda^{{\pm\nu}}\sum _{{k=0}}^{\infty}\frac{(\lambda^{2}-1)^{k}(\frac{1}{2}z)^{k}}{k!}\mathop{\mathscr{Z}_{{\nu\pm k}}\/}\nolimits\!\left(z\right),$ $|\lambda^{2}-1|<1$ .

Symbols:: $!$ : $n!$ : factorial, $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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
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If $\mathop{\mathscr{Z}\/}\nolimits=\mathop{I\/}\nolimits$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary.

¶ Examples

10.44.2

$\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\sum _{{k=0}}^{\infty}\frac{z^{k}}{k!}\mathop{J_{{\nu+k}}\/}\nolimits\!\left(z\right),$

$\!\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\sum _{{k=0}}^{\infty}(-1)^{k}\frac{z^{k}}{k!}\mathop{I_{{\nu+k}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $!$ : $n!$ : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $k$ : 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
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§10.44(ii) Addition Theorems

Notes:: Combine (10.23.2) and (10.27.1) with (10.27.6) or with (10.27.8).
Keywords:: modified Bessel functions
Referenced by:: §28.27
Permalink:: http://dlmf.nist.gov/10.44.ii

¶ Neumann’s Addition Theorem

Keywords:: Neumann’s addition theorem, modified Bessel functions

10.44.3 $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(u\pm v\right)=\sum _{{k=-\infty}}^{\infty}(\pm 1)^{k}\mathop{\mathscr{Z}_{{\nu+k}}\/}\nolimits\!\left(u\right)\mathop{I_{{k}}\/}\nolimits\!\left(v\right),$ $|v|<|u|$ .

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, $k$ : nonnegative integer and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.44.E3
Encodings:: TeX, pMML, png

The restriction $|v|<|u|$ is unnecessary when $\mathop{\mathscr{Z}\/}\nolimits=\mathop{I\/}\nolimits$ and $\nu$ is an integer.

¶ Graf’s and Gegenbauer’s Addition Theorems

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

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

Notes:: Combine (10.23.15)–(10.23.17) with (10.27.6), (10.27.8), and (10.4.3).
Keywords:: Neumann-type expansions, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.44.iii

10.44.4 $\left(\tfrac{1}{2}z\right)^{\nu}=\sum _{{k=0}}^{\infty}(-1)^{k}\frac{(\nu+2k)\mathop{\Gamma\/}\nolimits\!\left(\nu+k\right)}{k!}\mathop{I_{{\nu+2k}}\/}\nolimits\!\left(z\right),$ $\nu\neq 0,-1,-2,\ldots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.44.E4
Encodings:: TeX, pMML, png

10.44.5 $\mathop{K_{{0}}\/}\nolimits\!\left(z\right)=-\left(\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}z\right)+\EulerConstant\right)\mathop{I_{{0}}\/}\nolimits\!\left(z\right)+2\sum _{{k=1}}^{\infty}\frac{\mathop{I_{{2k}}\/}\nolimits\!\left(z\right)}{k},$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : 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
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10.44.6 $\mathop{K_{{n}}\/}\nolimits\!\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{{-n}}}{2}\sum _{{k=0}}^{{n-1}}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}\mathop{I_{{k}}\/}\nolimits\!\left(z\right)}{k!(n-k)}+(-1)^{{n-1}}\left(\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}z\right)-\mathop{\psi\/}\nolimits\!\left(n+1\right)\right)\mathop{I_{{n}}\/}\nolimits\!\left(z\right)+(-1)^{n}\sum _{{k=1}}^{\infty}\frac{(n+2k)\mathop{I_{{n+2k}}\/}\nolimits\!\left(z\right)}{k(n+k)},$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : 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
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where $\EulerConstant$ is Euler’s constant and $\mathop{\psi\/}\nolimits=\ifrac{{\mathop{\Gamma\/}\nolimits^{{\prime}}}}{\mathop{\Gamma\/}\nolimits}$ (§5.2).

§10.44(iv) Compendia

Keywords:: integrals of modified Bessel functions, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.44.iv

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).