10 Bessel Functions Spherical Bessel Functions 10.50 Wronskians and Cross-Products 10.52 Limiting Forms

§10.51 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/10.51
See also:: Annotations for Ch.10

§10.51(i) Unmodified Functions
§10.51(ii) Modified Functions

§10.51(i) Unmodified Functions

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Keywords:: derivatives, recurrence relations, spherical Bessel functions
Notes:: For (10.51.1) and (10.51.2) combine (10.6.1) and (10.6.2) with the definitions (10.47.3)–(10.47.5). For (10.51.3) apply induction with the aid of (10.51.2).
Permalink:: http://dlmf.nist.gov/10.51.i
See also:: Annotations for §10.51 and Ch.10

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

$f_{n-1}(z)+f_{n+1}(z)=((2n+1)/z)f_{n}(z),$

$nf_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_{n}^{\prime}(z),$ $n=1,2,\dots$ ,

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Symbols:: : integer, : 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

$f_{n}^{\prime}(z)=f_{n-1}(z)-((n+1)/z)f_{n}(z),$ $n=1,2,\dots$ ,

$f_{n}^{\prime}(z)=-f_{n+1}(z)+(n/z)f_{n}(z),$ $n=0,1,\dots.$

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Symbols:: : integer, : 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

$\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{n+1}f_{n}(z))=z^% {n-m+1}f_{n-m}(z),$ $m=0,1,\dots,n$ ,

$\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{-n}f_{n}(z))=(-1% )^{m}z^{-n-m}f_{n+m}(z),$ $m=0,1,\dots.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : integer, : integer, : 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

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Keywords:: derivatives, recurrence relations, spherical Bessel functions
Notes:: For (10.51.4) and (10.51.5) combine (10.29.1) and (10.29.2) with the definitions (10.47.7) and (10.47.9). For (10.51.6) apply induction with the aid of (10.51.5).
Permalink:: http://dlmf.nist.gov/10.51.ii
See also:: Annotations for §10.51 and Ch.10

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