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{{DISPLAYTITLE:Bessel Functions - 10.6 Recurrence Relations and Derivatives}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.6#E3X 10.6#E3X] || <math qid="Q3034">\displaystyle\BesselJ{0}'@{z} = -\BesselJ{1}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\BesselJ{0}'@{z} = -\BesselJ{1}@{z}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(1+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">diff( BesselJ(0, z), z$(1) ) = - BesselJ(1, z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">D[BesselJ[0, z], {z, 1}] == - BesselJ[1, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#E3X 10.6#E3X] || <math qid="Q3034">\displaystyle\BesselY{0}'@{z} = -\BesselY{1}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\BesselY{0}'@{z} = -\BesselY{1}@{z}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(1+k+1)} > 0, \realpart@@{((-0)+k+1)} > 0, \realpart@@{((-1)+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">diff( BesselY(0, z), z$(1) ) = - BesselY(1, z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">D[BesselY[0, z], {z, 1}] == - BesselY[1, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#E3Xa 10.6#E3Xa] || <math qid="Q3036">\displaystyle\HankelH{1}{0}'@{z} = -\HankelH{1}{1}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\HankelH{1}{0}'@{z} = -\HankelH{1}{1}@{z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">diff( HankelH1(0, z), z$(1) ) = - HankelH1(1, z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">D[HankelH1[0, z], {z, 1}] == - HankelH1[1, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#E3Xa 10.6#E3Xa] || <math qid="Q3036">\displaystyle\HankelH{2}{0}'@{z} = -\HankelH{2}{1}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\HankelH{2}{0}'@{z} = -\HankelH{2}{1}@{z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">diff( HankelH2(0, z), z$(1) ) = - HankelH2(1, z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">D[HankelH2[0, z], {z, 1}] == - HankelH2[1, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#Ex5 10.6#Ex5] || <math qid="Q3037">f_{\nu-1}(z)+f_{\nu+1}(z) = (2\nu/\lambda)z^{-q}f_{\nu}(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{\nu-1}(z)+f_{\nu+1}(z) = (2\nu/\lambda)z^{-q}f_{\nu}(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[nu - 1](z)+ f[nu + 1](z) = (2*nu/lambda)*(z)^(- q)* f[nu](z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, \[Nu]- 1][z]+ Subscript[f, \[Nu]+ 1][z] == (2*\[Nu]/\[Lambda])*(z)^(- q)* Subscript[f, \[Nu]][z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#Ex15 10.6#Ex15] || <math qid="Q3048">p_{\nu+1}-p_{\nu-1} = -\frac{2\nu}{a}q_{\nu}-\frac{2\nu}{b}r_{\nu}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu+1}-p_{\nu-1} = -\frac{2\nu}{a}q_{\nu}-\frac{2\nu}{b}r_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu + 1]- p[nu - 1] = -(2*nu)/(a)*q[nu]-(2*nu)/(b)*r[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]+ 1]- Subscript[p, \[Nu]- 1] == -Divide[2*\[Nu],a]*Subscript[q, \[Nu]]-Divide[2*\[Nu],b]*Subscript[r, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#Ex16 10.6#Ex16] || <math qid="Q3049">q_{\nu+1}+r_{\nu} = \frac{\nu}{a}p_{\nu}-\frac{\nu+1}{b}p_{\nu+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>q_{\nu+1}+r_{\nu} = \frac{\nu}{a}p_{\nu}-\frac{\nu+1}{b}p_{\nu+1}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">q[nu + 1]+ r[nu] = (nu)/(a)*p[nu]-(nu + 1)/(b)*p[nu + 1]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[q, \[Nu]+ 1]+ Subscript[r, \[Nu]] == Divide[\[Nu],a]*Subscript[p, \[Nu]]-Divide[\[Nu]+ 1,b]*Subscript[p, \[Nu]+ 1]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#Ex17 10.6#Ex17] || <math qid="Q3050">r_{\nu+1}+q_{\nu} = \frac{\nu}{b}p_{\nu}-\frac{\nu+1}{a}p_{\nu+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r_{\nu+1}+q_{\nu} = \frac{\nu}{b}p_{\nu}-\frac{\nu+1}{a}p_{\nu+1}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">r[nu + 1]+ q[nu] = (nu)/(b)*p[nu]-(nu + 1)/(a)*p[nu + 1]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[r, \[Nu]+ 1]+ Subscript[q, \[Nu]] == Divide[\[Nu],b]*Subscript[p, \[Nu]]-Divide[\[Nu]+ 1,a]*Subscript[p, \[Nu]+ 1]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6#Ex18 10.6#Ex18] || <math qid="Q3051">s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-\frac{\nu^{2}}{ab}p_{\nu}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-\frac{\nu^{2}}{ab}p_{\nu}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(diff( BesselJ(nu, a), a$(1) )*diff( BesselY(nu, b), b$(1) )- diff( BesselJ(nu, b), b$(1) )*diff( BesselY(nu, a), a$(1) )) = (1)/(2)*p[nu + 1]+(1)/(2)*p[nu - 1]-((nu)^(2))/(a*b)*p[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(D[BesselJ[\[Nu], a], {a, 1}]*D[BesselY[\[Nu], b], {b, 1}]- D[BesselJ[\[Nu], b], {b, 1}]*D[BesselY[\[Nu], a], {a, 1}]) == Divide[1,2]*Subscript[p, \[Nu]+ 1]+Divide[1,2]*Subscript[p, \[Nu]- 1]-Divide[\[Nu]^(2),a*b]*Subscript[p, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.6.E10 10.6.E10] || <math qid="Q3052">p_{\nu}s_{\nu}-q_{\nu}r_{\nu} = 4/(\pi^{2}ab)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu}s_{\nu}-q_{\nu}r_{\nu} = 4/(\pi^{2}ab)</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu]*(diff( BesselJ(nu, a), a$(1) )*diff( BesselY(nu, b), b$(1) )- diff( BesselJ(nu, b), b$(1) )*diff( BesselY(nu, a), a$(1) ))- q[nu]*r[nu] = 4/((Pi)^(2)* a*b)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]]*(D[BesselJ[\[Nu], a], {a, 1}]*D[BesselY[\[Nu], b], {b, 1}]- D[BesselJ[\[Nu], b], {b, 1}]*D[BesselY[\[Nu], a], {a, 1}])- Subscript[q, \[Nu]]*Subscript[r, \[Nu]] == 4/((Pi)^(2)* a*b)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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