§10.51 Recurrence Relations and Derivatives

Let $f_n(z)$ denote any of ${\mathsf{j}}_n\left(z\right)$, ${\mathsf{y}}_n\left(z\right)$, ${\mathsf{h}}_n^{(1)}\left(z\right)$, or ${\mathsf{h}}_n^{(2)}\left(z\right)$. Then

10.51.1		$f_{n-1}(z)+f_{n+1}(z)$	$=((2n+1)/z)f_n(z),$
		$n{f}_{n-1}(z)-(n+1)f_{n+1}(z)$	$=(2n+1)f_n^{\prime }(z),$
	$n=1,2,\mathrm{\dots }$,
	ⓘ Symbols: $n$: integer, $z$: complex variable and $f_n(z)$: a spherical Bessel function A&S Ref: 10.1.19, 10.1.20 Referenced by: §10.50, §10.51(i), §10.56 Permalink: http://dlmf.nist.gov/10.51.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(i), §10.51 and Ch.10

10.51.2		$f_n^{\prime }(z)$	$=f_{n-1}(z)-((n+1)/z)f_n(z),$
	$n=1,2,\mathrm{\dots }$,
		$f_n^{\prime }(z)$	$=-f_{n+1}(z)+(n/z)f_n(z),$
	$n=0,1,\mathrm{\dots }.$
ⓘ Symbols: $n$: integer, $z$: complex variable and $f_n(z)$: a spherical Bessel function A&S Ref: 10.1.21, 10.1.22 Referenced by: §10.50, §10.51(i) Permalink: http://dlmf.nist.gov/10.51.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(i), §10.51 and Ch.10

10.51.3		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^m(z^{n+1}{f}_n(z))$	$=z^{n-m+1}{f}_{n-m}(z),$
	$m=0,1,\mathrm{\dots },n$,
		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^m(z^{-n}{f}_n(z))$	$={(-1)}^m{z}^{-n-m}{f}_{n+m}(z),$
	$m=0,1,\mathrm{\dots }.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $m$: integer, $n$: integer, $z$: complex variable and $f_n(z)$: a spherical Bessel function A&S Ref: 10.1.23, 10.1.24 Referenced by: §10.49(iii), §10.51(i) Permalink: http://dlmf.nist.gov/10.51.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(i), §10.51 and Ch.10

Let $g_n(z)$ denote ${\mathsf{i}}_n^{(1)}\left(z\right)$, ${\mathsf{i}}_n^{(2)}\left(z\right)$, or ${(-1)}^n$ ${\mathsf{k}}_n\left(z\right)$. Then

10.51.4		$g_{n-1}(z)-g_{n+1}(z)$	$=((2n+1)/z)g_n(z)$
		$n{g}_{n-1}(z)+(n+1)g_{n+1}(z)$	$=(2n+1)g_n^{\prime }(z),$
	$n=1,2,\mathrm{\dots }$,
	ⓘ Symbols: $n$: integer, $z$: complex variable and $g_n(z)$: a spherical Bessel function A&S Ref: 10.2.18, 10.2.19 Referenced by: §10.51(ii) Permalink: http://dlmf.nist.gov/10.51.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(ii), §10.51 and Ch.10

10.51.5		$g_n^{\prime }(z)$	$=g_{n-1}(z)-((n+1)/z)g_n(z),$
	$n=1,2,\mathrm{\dots }$,
		$g_n^{\prime }(z)$	$=g_{n+1}(z)+(n/z)g_n(z),$
	$n=0,1,\mathrm{\dots }.$
ⓘ Symbols: $n$: integer, $z$: complex variable and $g_n(z)$: a spherical Bessel function A&S Ref: 10.2.20, 10.2.21 Referenced by: §10.51(ii) Permalink: http://dlmf.nist.gov/10.51.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(ii), §10.51 and Ch.10

10.51.6		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^m(z^{n+1}{g}_n(z))$	$=z^{n-m+1}{g}_{n-m}(z),$
	$m=0,1,\mathrm{\dots },n$,
		${\left({\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}\right)}^m(z^{-n}{g}_n(z))$	$=z^{-n-m}{g}_{n+m}(z),$
	$m=0,1,\mathrm{\dots }.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $m$: integer, $n$: integer, $z$: complex variable and $g_n(z)$: a spherical Bessel function A&S Ref: 10.2.22, 10.2.23 Referenced by: §10.51(ii) Permalink: http://dlmf.nist.gov/10.51.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.51(ii), §10.51 and Ch.10