Editing Bessel function (section)

=== Recurrence relations ===
The functions {{mvar|J<sub>α</sub>}}, {{mvar|Y<sub>α</sub>}}, {{math|''H''{{su|b=''α''|p=(1)}}}}, and {{math|''H''{{su|b=''α''|p=(2)}}}} all satisfy the [[recurrence relation]]s<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_361.htm p. 361, 9.1.27].</ref>
<math display="block">\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)</math>
and
<math display="block"> 2\frac{dZ_\alpha (x)}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x),</math>
where {{mvar|Z}} denotes {{mvar|J}}, {{mvar|Y}}, {{math|''H''<sup>(1)</sup>}}, or {{math|''H''<sup>(2)</sup>}}. These two identities are often combined, e.g. added or subtracted, to yield various other relations. In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives). In particular, it follows that<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_361.htm p. 361, 9.1.30].</ref>
<math display="block">\begin{align}
\left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ x^\alpha Z_\alpha (x) \right] &= x^{\alpha - m} Z_{\alpha - m} (x), \\
\left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] &= (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}.
\end{align}</math>

''Modified'' Bessel functions follow similar relations:
<math display="block">e^{\left(\frac{x}{2}\right)\left(t+\frac{1}{t}\right)} = \sum_{n=-\infty}^\infty I_n(x) t^n</math>
and
<math display="block">e^{z \cos \theta} = I_0(z) + 2\sum_{n=1}^\infty I_n(z) \cos n\theta</math>
and
<math display="block"> \frac{1}{2\pi} \int_0^{2\pi} e^{z \cos (m\theta) + y \cos \theta} d\theta = I_0(z)I_0(y) + 2\sum_{n=1}^\infty I_n(z)I_{mn}(y).</math>

The recurrence relation reads
<math display="block">\begin{align}
C_{\alpha-1}(x) - C_{\alpha+1}(x) &= \frac{2\alpha}{x} C_\alpha(x), \\[1ex]
C_{\alpha-1}(x) + C_{\alpha+1}(x) &= 2\frac{d}{dx}C_\alpha(x),
\end{align}</math>
where {{mvar|C<sub>α</sub>}} denotes {{mvar|I<sub>α</sub>}} or {{math|''e''<sup>''αi''π</sup>''K<sub>α</sub>''}}. These recurrence relations are useful for discrete diffusion problems.