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10 Bessel FunctionsSpherical Bessel Functions

§10.51 Recurrence Relations and Derivatives

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§10.51(i) Unmodified Functions

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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).
Keywords:
derivatives, spherical Bessel functions
Permalink:
http://dlmf.nist.gov/10.51.i

Let f_{n}(z) denote any of \mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right), \mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right), \mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right), or \mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\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:
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
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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:
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:
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§10.51(ii) Modified Functions

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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).
Keywords:
derivatives, spherical Bessel functions
Permalink:
http://dlmf.nist.gov/10.51.ii

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

10.51.4
g_{{n-1}}(z)-g_{{n+1}}(z)=((2n+1)/z)g_{n}(z)
ng_{{n-1}}(z)+(n+1)g_{{n+1}}(z)=(2n+1)g_{n}^{{\prime}}(z),n=1,2,\ldots,
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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:
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10.51.5
g_{n}^{{\prime}}(z)=g_{{n-1}}(z)-((n+1)/z)g_{n}(z),n=1,2,\ldots,
g_{n}^{{\prime}}(z)=g_{{n+1}}(z)+(n/z)g_{n}(z),n=0,1,\ldots.
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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:
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10.51.6
\left(\frac{1}{z}\frac{d}{dz}\right)^{m}(z^{{n+1}}g_{n}(z))=z^{{n-m+1}}g_{{n-m% }}(z),m=0,1,\ldots,n,
\left(\frac{1}{z}\frac{d}{dz}\right)^{m}(z^{{-n}}g_{n}(z))=z^{{-n-m}}g_{{n+m}}% (z),m=0,1,\ldots.
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