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I am curious if there is a closed form that represents the coefficients of the inverse of the modified Bessel function of the first kind $I_{0}(x)$. I can find the series representation using InverseSeries[Series[BesselI[0,x],{x,0,20}]] in Mathematica.

$$I^{-1}_{0}(x)=\sum^{\infty}_{n=1}{(-1)^{n+1}a_{n}(x-1)^{n-\frac{1}{2}}}$$

The first few coefficients $a_{n}$ are: $2$, $\frac{1}{4}$, $\frac{47}{576}$, $\frac{161}{4608}$, $\frac{565571}{33177600}$, $...$ Or the series can be expressed as: $$I^{-1}_{0}(x)=2\sqrt{x-1}\sum^{\infty}_{n=0}{(-1)^{n}b_{n}(x-1)^{n}}$$

In this case, the coefficients would just be divided by two: $1$, $\frac{1}{8}$, $\frac{47}{1152}$, $\frac{161}{9216}$, $\frac{565571}{66355200}$, $...$ Other than the coefficients, the inverse Bessel function is very similar to the $\cosh^{-1}(x)$ series. If there is no closed form, or neat closed form for either of the coefficients $a_{n}$ or $b_{n}$, is there a better way, or more condensed way to represent these sums?

Gary
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Kyler Rusin
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  • A similar question is from here. Inverses of most special functions usually have complicated forms. However, you can find a recurrence relation. You could also represent the coefficients as an $n$th derivative, like in a Taylor series, but that may not be satisfying – Тyma Gaidash Mar 28 '24 at 23:23

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