# §10.51 Recurrence Relations and Derivatives

## §10.51(i) Unmodified Functions

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

 10.51.1 $\displaystyle f_{n-1}(z)+f_{n+1}(z)$ $\displaystyle=((2n+1)/z)f_{n}(z),$ $\displaystyle nf_{n-1}(z)-(n+1)f_{n+1}(z)$ $\displaystyle=(2n+1)f_{n}^{\prime}(z),$ $n=1,2,\dots$, ⓘ 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 $\displaystyle f_{n}^{\prime}(z)$ $\displaystyle=f_{n-1}(z)-((n+1)/z)f_{n}(z),$ $n=1,2,\dots$, $\displaystyle f_{n}^{\prime}(z)$ $\displaystyle=-f_{n+1}(z)+(n/z)f_{n}(z),$ $n=0,1,\dots.$ ⓘ 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 $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{n+1% }f_{n}(z))$ $\displaystyle=z^{n-m+1}f_{n-m}(z),$ $m=0,1,\dots,n$, $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{-n}% f_{n}(z))$ $\displaystyle=(-1)^{m}z^{-n-m}f_{n+m}(z),$ $m=0,1,\dots.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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

## §10.51(ii) Modified Functions

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

 10.51.4 $\displaystyle g_{n-1}(z)-g_{n+1}(z)$ $\displaystyle=((2n+1)/z)g_{n}(z)$ $\displaystyle ng_{n-1}(z)+(n+1)g_{n+1}(z)$ $\displaystyle=(2n+1)g_{n}^{\prime}(z),$ $n=1,2,\dotsc$, ⓘ 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 $\displaystyle g_{n}^{\prime}(z)$ $\displaystyle=g_{n-1}(z)-((n+1)/z)g_{n}(z),$ $n=1,2,\dotsc$, $\displaystyle g_{n}^{\prime}(z)$ $\displaystyle=g_{n+1}(z)+(n/z)g_{n}(z),$ $n=0,1,\dotsc.$ ⓘ 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 $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{n+1% }g_{n}(z))$ $\displaystyle=z^{n-m+1}g_{n-m}(z),$ $m=0,1,\dotsc,n$, $\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{-n}% g_{n}(z))$ $\displaystyle=z^{-n-m}g_{n+m}(z),$ $m=0,1,\dotsc.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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