How to get nth derivative of $\arcsin x$

Question

I want to calculate the nth derivative of $\arcsin x$. I know $$ \frac{d}{dx}\arcsin x=\frac1{\sqrt{1-x^2}} $$ And $$ \frac{d^n}{dx^n} \frac1{\sqrt{1-x^2}} = \frac{d}{dx} \left(P_{n-1}(x) \frac1{\sqrt{1-x^2}}\right) = \left(-\frac{x}{(1-x^2)^{}} P_{n-1}(x) + \frac{dP_{n-1}}{dx}\right)\frac1{\sqrt{1-x^2}} = P_n(x) \frac1{\sqrt{1-x^2}} $$ Hence we have the recursive relation of $P_n$: $$ P_{n}(x)=-\frac{x}{(1-x^2)^{}} P_{n-1}(x) + \frac{dP_{n-1}}{dx}, \:P_0(x) = 1 $$ My question is how to solve the recursive relation involving function and derivative. I think it should use the generating function, but not sure what it is.

Answer 1

Let $~P_n(x)~=~\dfrac{2^n}{n!}~\Big(\sqrt{1-x^2}\Big)^{2n+1}~\bigg(\dfrac1{\sqrt{1-x^2}}\bigg)^{(n)}.~$ Then its coefficients form the sequence described here.

Answer 2

Outline of a proof that does not use the recursive relation but does yield the correct answer.

Use Faà di Bruno's formula for $n$-fold derivatives in it's "Bell polynomial" form. This omits the question how the tuples affect the final form of the derivatives, so start with:

\begin{equation} \frac{d^n}{dx^n}f(g(x)) = \sum_{k=1}^n f^{(n)}(g(x)) \cdot B_{n,k} (g'(x),g^{(2)}(x),\cdots) \end{equation} with \begin{equation} B_{n,k}(x_1,x_2,\cdots) = \sum_{\sum_i j_i = k, \sum_\ell \ell j_\ell = n} \frac{n!}{\prod_i j_i!} \prod_i\left(\frac{x_i}{i!}\right)^{j_i} \end{equation}

Once this is done, it's useful to think about how the tuples in the Bell's polynomial summation are constrained (hint: only 1 term survives).

The final answer you're looking for is:

\begin{equation} \frac{d^n (1/\sqrt{1-x^2})}{d x^n} = \sum_{k=1}^n (-)^k \left(\frac12 - k\right)_k \left(1-x^2\right)^{-1/2-k} \frac{(2k-n+1)_{2(n-k)} (2x)^{2k-n}}{(n-k)!} \end{equation} where the subbed expressions are Pochhammer symbols.

I think that relating this answer to the expression derived when say $n=4$ is taken or $n=7$ or whatever, you will find that the combinatorial expressions from the explicit $n$ add up in non-intuitive ways to the expression found here, and that in that sense, the setup with the recursive approach should be reconsidered.

Hope someone will find this useful.

Answer 3

Let $v=v(x)=1-x^2$. For $k\ge0$, making use of the Faa di Bruno formula and some properties of the partial Bell polynomials $B_{n,k}$, we have \begin{align*} \biggl[\frac{1}{\sqrt{1-x^2}\,}\biggr]^{(k)} &=\sum_{j=0}^{k}\frac{\operatorname{d}^j}{\operatorname{d} v^j}\biggl(\frac{1}{\sqrt{v}\,}\biggr) B_{k,j}(-2x,-2,0,\dotsc,0)\\ &=\sum_{j=0}^{k}\biggl\langle-\frac12\biggr\rangle_j\frac{1}{v^{1/2+j}} (-1)^j2^jB_{k,j}(x,1,0,\dotsc,0)\\ &=\sum_{j=0}^{k}\biggl\langle-\frac12\biggr\rangle_j \frac{(-1)^j}{(1-x^2)^{1/2+j}} 2^j \frac{1}{2^{k-j}}\frac{k!}{j!}\binom{j}{k-j}x^{2j-k}\\ &=\frac1{\sqrt{1-x^2}\,}\frac{k!}{2^k} \frac1{x^k}\sum_{j=0}^{k}\frac{2^{j}(2j-1)!!}{j!}\binom{j}{k-j}\frac{x^{2j}}{(1-x^2)^{j}}. \end{align*} Consequently, we acquire \begin{equation} (\arcsin x)^{(k+1)}=\frac1{\sqrt{1-x^2}\,}\frac{k!}{2^k} \frac1{x^k}\sum_{j=0}^{k}\frac{2^{j}(2j-1)!!}{j!}\binom{j}{k-j}\frac{x^{2j}}{(1-x^2)^{j}}, \quad k\ge0. \end{equation} Equivalently, \begin{equation} (\arcsin x)^{(n)}=\frac1{\sqrt{1-x^2}\,}\frac{(n-1)!}{(2x)^{n-1}} \sum_{j=0}^{n-1}\frac{2^{j}(2j-1)!!}{j!}\binom{j}{n-j-1}\frac{x^{2j}}{(1-x^2)^{j}}, \quad n\ge1. \end{equation} In conclusion, we acquire \begin{equation} P_n(x)=\frac{n!}{(2x)^n} \sum_{j=0}^{n}\frac{2^{j}(2j-1)!!}{j!}\binom{j}{n-j}\frac{x^{2j}}{(1-x^2)^{j}}, \quad n\ge0. \end{equation}

References

1. Siqintuya Jin, Bai-Ni Guo, and Feng Qi, Partial Bell polynomials, falling and rising factorials, Stirling numbers, and combinatorial identities, Computer Modeling in Engineering & Sciences 132 (2022), no. 3, 781--799; available online at https://doi.org/10.32604/cmes.2022.019941.
2. F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1.
3. F. Qi, D.-W. Niu, D. Lim, and Y.-H. Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, J. Math. Anal. Appl. 491 (2020), no. 2, Article 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382.