28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.23 Expansions in Series of Bessel Functions 28.25 Asymptotic Expansions for Large

\Re z

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

ⓘ

Keywords:: cross-products of Bessel functions and modified Bessel functions, expansions in series of, expansions in series of cross-products of Bessel functions and modified Bessel functions, modified Mathieu functions, radial Mathieu functions
Notes:: See Meixner and Schäfke (1954, §2.64 and p. 201).
Referenced by:: Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/28.24
Addition (effective with 1.0.5):: The reference to Larsen et al. (2009) has been added at the end of this subsection.
See also:: Annotations for Ch.28

Throughout this section $\varepsilon_{0}=2$ and $\varepsilon_{s}=1$ , $s=1,2,3,\dotsc$ .

With ${\cal C}_{\mu}^{(j)}$ , $c^{\nu}_{n}(q)$ , $A_{n}^{m}(q)$ , and $B_{n}^{m}(q)$ as in §28.23,

28.24.1		$c_{2n}^{\nu}(h^{2}){\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)=\sum_{\ell=-% \infty}^{\infty}(-1)^{\ell}c_{2\ell}^{\nu}(h^{2})J_{\ell-n}\left(he^{-z}\right% )\mathcal{C}_{\nu+n+\ell}^{(j)}(he^{z}),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $n$ : integer, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $c_{2m}(q)$ : coefficients and $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions A&S Ref: 20.4.12 Referenced by: §28.22(ii), §28.24, item (a) Permalink: http://dlmf.nist.gov/28.24.E1 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28

where $j=1,2,3,4$ and $n\in\mathbb{Z}$ .

In the case when $\nu$ is an integer,

28.24.2	$\displaystyle\varepsilon_{s}{\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell}^{2m}(h^% {2})}{A_{2s}^{2m}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s}% ^{(j)}(he^{z})+J_{\ell+s}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})% \right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions, $\varepsilon_{s}$ : constants and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.6.7 Permalink: http://dlmf.nist.gov/28.24.E2 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.3	$\displaystyle{\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1% }(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_% {\ell+s+1}^{(j)}(he^{z})+J_{\ell+s+1}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j% )}(he^{z})\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.6.8 Permalink: http://dlmf.nist.gov/28.24.E3 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28

28.24.4	$\displaystyle{\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{B_{2\ell+1}^{2m+1% }(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_% {\ell+s+1}^{(j)}(he^{z})-J_{\ell+s+1}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j% )}(he^{z})\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $B_{m}(q)$ : Fourier coefficient A&S Ref: 20.6.9 Permalink: http://dlmf.nist.gov/28.24.E4 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.5	$\displaystyle{\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{B_{2\ell+2}^{2m+2% }(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_% {\ell+s+2}^{(j)}(he^{z})-J_{\ell+s+2}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j% )}(he^{z})\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $B_{m}(q)$ : Fourier coefficient A&S Ref: 20.6.10 Permalink: http://dlmf.nist.gov/28.24.E5 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28

where $j=1,2,3,4$ , and $s=0,1,2,\dots$ .

Also, with $I_{n}$ and $K_{n}$ denoting the modified Bessel functions (§10.25(ii)), and again with $s=0,1,2,\dots$ ,

28.24.6	$\displaystyle\varepsilon_{s}\operatorname{Ie}_{2m}\left(z,h\right)$	$\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\dfrac{A_{2\ell}^{2m}(h% ^{2})}{A_{2s}^{2m}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+s}\left(% he^{z}\right)+I_{\ell+s}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}\right)% \right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{Ie}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\varepsilon_{s}$ : constants and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.8.8 Permalink: http://dlmf.nist.gov/28.24.E6 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.7	$\displaystyle\operatorname{Io}_{2m+2}\left(z,h\right)$	$\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\dfrac{B_{2\ell+2}^{2m+% 2}(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+% s+2}\left(he^{z}\right)-I_{\ell+s+2}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}% \right)\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{Io}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.24.E7 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.8	$\displaystyle\operatorname{Ie}_{2m+1}\left(z,h\right)$	$\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\dfrac{B_{2\ell+1}^{2m+% 1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+% s+1}\left(he^{z}\right)+I_{\ell+s+1}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}% \right)\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{Ie}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.24.E8 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.9	$\displaystyle\operatorname{Io}_{2m+1}\left(z,h\right)$	$\displaystyle=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1% }(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+s% +1}\left(he^{z}\right)-I_{\ell+s+1}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}% \right)\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{Io}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.24.E9 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.10	$\displaystyle\varepsilon_{s}\operatorname{Ke}_{2m}\left(z,h\right)$	$\displaystyle=\sum_{\ell=0}^{\infty}\frac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h% ^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s}\left(he^{z}\right)+I_{% \ell+s}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{Ke}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\varepsilon_{s}$ : constants and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.8.9 Permalink: http://dlmf.nist.gov/28.24.E10 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.11	$\displaystyle\operatorname{Ko}_{2m+2}\left(z,h\right)$	$\displaystyle=\sum_{\ell=0}^{\infty}\frac{B_{2\ell+2}^{2m+2}(h^{2})}{B_{2s+2}^% {2m+2}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s+2}\left(he^{z}% \right)-I_{\ell+s+2}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{Ko}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.24.E11 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.12	$\displaystyle\operatorname{Ke}_{2m+1}\left(z,h\right)$	$\displaystyle=\sum_{\ell=0}^{\infty}\frac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^% {2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s+1}\left(he^{z}% \right)-I_{\ell+s+1}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{Ke}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.24.E12 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28
28.24.13	$\displaystyle\operatorname{Ko}_{2m+1}\left(z,h\right)$	$\displaystyle=\sum_{\ell=0}^{\infty}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^% {2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_{\ell+s+1}\left(he^{z}% \right)+I_{\ell+s+1}\left(he^{-z}\right)K_{\ell-s}\left(he^{z}\right)\right).$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{Ko}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient Referenced by: §28.24, item (a) Permalink: http://dlmf.nist.gov/28.24.E13 Encodings: TeX, pMML, png See also: Annotations for §28.24 and Ch.28

The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the $z$ -plane.

For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

28.23 Expansions in Series of Bessel Functions 28.25 Asymptotic Expansions for Large

\Re z

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