$n$-th derivative of $\exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$

Question

Let $\lambda$ and $\mu$ be two positive real numbers and let denote $f$ the function defined as:

$$\forall x>0,~f(x):= \exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right).$$

I am struggling to find out an explicit expression of $\partial_x^n f$. According to this answer, it seems that:

$$\partial_x^n f(x) = f(x)\frac{P_{\lambda, \mu}(x)}{2^n \mu^{2n}x^{2n}}$$

where $P_{\lambda, \mu}$ is a polynomial function of $x$ depending on $\lambda$ and $\mu$.

Does anyone know how to deal with this derivative ?

Any hint, solution or reference will be highly appreciated!

Thank you.

Answer 1

$\newcommand\la\lambda$Let $$g(x):=-\frac\la{2\mu^2}\frac{(x-\mu)^2}x =-\frac\la{2\mu^2}(x-2\mu+\mu^2/x)$$ so that $$f(x)=e^{g(x)}.$$ So, by the Faà di Bruno formula, $$f^{(n)}(x)=f(x)\sum_{\pi\in\Pi_n}\prod_{B\in\pi}g^{(|B|)}(x),$$ where $\Pi_n$ is the set of all partitions of the set $\{1,\dots,n\}$, $|B|$ is the cardinality of $B$, and $$g^{(k)}(x)=-\frac\la{2\mu^2}\,1(k=1)-\frac\la2\,(-1)^k k!x^{-k-1}$$ for $k=1,2,\dots$.

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For $y_n:=f^{(n)}(x)/n!$, Mathematica gives the recursion $$y(n+2) \left(-\lambda \mu ^2+4 \mu ^2 n x+\lambda x^2+8 \mu ^2 x\right)+2 \mu ^2 (n+3) x^2 y(n+3)+2 y(n+1) \left(\mu ^2+\mu ^2 n+\lambda x\right)+\lambda y(n)=0$$ for $n\ge0$, which implies that $$f^{(n)}(x)=f(x)n!\frac{P_n(x)}{2^n \mu^{2n}x^{2n}},$$ where, for each $n$, $P_n(x)=P_{n;\la,\mu}(x)$ is a polynomial such that $$P_0(x)=1,\quad P_1(x)=\lambda \left(\mu ^2-x^2\right),\quad P_2(x)=\frac{1}{2} \lambda \left(\lambda \mu ^4+\lambda x^4-2 \lambda \mu ^2 x^2-4 \mu ^4 x\right),$$ and the recurrence $$4 \lambda \mu ^4 x^4 P_n(x)+4 \mu ^2 x^2 P_{n+1}(x) \left(\mu ^2 (n+1)+\lambda x\right)+P_{n+2}(x) \left(-\lambda \mu ^2+4 \mu ^2 (n+2) x+\lambda x^2\right)+(n+3) P_{n+3}(x)=0$$ holds for $n\ge0$.

Answer 2

The polynomials $P$ are related to the generalized Laguerre polynomials, see A066667. After some algebra, the $n$-th derivative of $f(x)$ can be written as \begin{align}\label{eq:1}\tag{1} \frac{\partial^n_x f(x)}{f(x)}= \left(-\frac{\lambda }{2\mu ^2}\right)^n + \sum_{k=1}^n \frac{n!}{(n-k)!} \, \frac{(-1)^k \mu ^2 }{k \, x^{k+1}} \left(-\frac{\lambda }{2\mu^2}\right)^{n-k+1} L_{k-1}^{(1)}\left(\frac{\lambda }{2 x}\right), \end{align} with the generalized Laguerre polynomials $L^{(\alpha)}_n(x)$.

Answer 3

$\newcommand\la\lambda$Here is a very simple expression for $f^{(N)}(x)$:

Note that $$f(x)=g(x)h(bx),$$ where $$g(x):=e^{a(x-2\mu)},\quad h(u):=e^{1/u},\quad a:=-\frac\la{2\mu^2},\quad b:=-\frac2\la.$$ So, by the Leibniz differentiation formula, $$f^{(N)}(x)=g(x)\sum_{n=0}^N\binom Nn a^{N-n} b^n h^{(n)}(bx).$$ In turn, it is easily checked by induction on $n$ that $$h^{(n)}(u)=h(u)\sum_{k=0}^n c_{n,k}u^{-k-n}, \tag{10}\label{10}$$ where $$c_{n,k}:=(-1)^n\frac{n!}{k!}\binom{n-1}{k-1}$$ (with $\binom{n-1}{k-1}=0$ for $k=0$).

So, $$f^{(N)}(x)=f(x)\sum_{n=0}^N\binom Nn a^{N-n} b^n \sum_{k=0}^n (-1)^n\frac{n!}{k!}\binom{n-1}{k-1}(bx)^{-k-n}.$$

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Note that, in view of the partial fraction decomposition, \eqref{10} allows one to get comparatively simple expressions for the $n$th derivative of any expression of the form $e^{P(x)/Q(x)}$, where $P(x)$ is any complex polynomial and $Q(x)$ is any complex polynomial without multiple roots. Moreover, the no-multiple-roots restriction can be removed, because any complex polynomial with multiple roots and its derivatives can be approximated however closely by a complex polynomial without multiple roots and its derivatives.

Also, writing (say) $\sin\frac1x=\frac1{2i}(e^{-1/(ix)}-e^{1/(ix)})$, we can use \eqref{10} to get a rather simple expression for the $n$th derivative of $\sin\frac1x$.

Answer 4

It is easy to see that we can write \begin{equation*} h(x)=\ln f(x)=-\frac{\lambda(x-\mu)^2}{2\mu^2x}=\frac{\lambda}{2\mu^2}\biggl(2\mu-x-\frac{\mu^2}{x}\biggr). \end{equation*} Hence, its derivatives are \begin{equation*} h'(x)=-\frac{\lambda}{2\mu^2}\biggl(1-\frac{\mu^2}{x^2}\biggr) =\frac{\lambda}{2}\biggl(\frac{1}{x^2}-\frac{1}{\mu^2}\biggr) \end{equation*} and \begin{equation*} h^{(k+1)}(x)=(-1)^{k}\frac{\lambda}{2}\frac{(k+1)!}{x^{k+2}},\quad k\in\mathbb{N}. \end{equation*} Making use of the identity \begin{equation*} B_{n,k}(x_1,x_2,\dotsc,x_{n-k+1}) =\frac{n!}{(n-k)!}\sum_{\ell=0}^{k}\frac{x_1^\ell}{\ell!} B_{n-k,k-\ell}\biggl(\frac{x_2}{2}, \frac{x_3}{3},\dotsc,\frac{x_{n+\ell-2k+2}}{n+\ell-2k+2}\biggr) \end{equation*} for $n\ge k\ge0$, where $B_{n,k}$ stands for the partial Bell polynomials (also known as the Bell polynomials of the second kind), we obtain \begin{align*} &\quad B_{n,k}\bigl(h'(x),h''(x),\dotsc,h^{(n-k+1)}(x)\bigr)\\ &=\frac{n!}{(n-k)!}\sum_{\ell=0}^{k}\frac{[h'(x)]^\ell}{\ell!} B_{n-k,k-\ell}\biggl(\frac{h''(x)}{2}, \frac{h'''(x)}{3}, \dotsc, \frac{h^{(n+\ell-2k+2)}(x)}{n+\ell-2k+2}\biggr)\\ &=\frac{n!}{(n-k)!}\sum_{\ell=0}^{k}\frac{1}{\ell!} \biggl[\frac{\lambda}{2}\biggl(\frac{1}{x^2}-\frac{1}{\mu^2}\biggr)\biggr]^\ell B_{n-k,k-\ell}\biggl(-\frac{1}{2}\frac{\lambda}{2}\frac{2!}{x^{3}}, \frac{1}{3} \frac{\lambda}{2}\frac{3!}{x^{4}}, \dotsc, \frac{(-1)^{n+\ell-2k+1}}{n+\ell-2k+2}\frac{\lambda}{2}\frac{(n+\ell-2k+2)!}{x^{n+\ell-2k+3}}\biggr)\\ &=\frac{n!}{(n-k)!}\sum_{\ell=0}^{k}\frac{1}{\ell!} \biggl[\frac{\lambda}{2}\biggl(\frac{1}{x^2}-\frac{1}{\mu^2}\biggr)\biggr]^\ell \biggl(\frac{\lambda}{2}\biggr)^{k-\ell} \frac{(-1)^{n-k}}{x^{n+k-2\ell}} B_{n-k,k-\ell}(1!, 2!, \dotsc,(n+\ell-2k+1)!)\\ &=\frac{(-1)^{n-k}}{(n-k)!}\frac{n!}{x^{n+k}}\biggl(\frac{\lambda}{2}\biggr)^{k} \sum_{\ell=0}^{k}\frac{1}{\ell!}\biggl(1-\frac{x^2}{\mu^2}\biggr)^\ell\binom{n-k-1}{k-\ell-1}\frac{(n-k)!}{(k-\ell)!}\\ &=\frac{(-1)^{n+k}}{x^{n+k}}\frac{n!}{k!}\biggl(\frac{\lambda}{2}\biggr)^{k} \sum_{\ell=0}^{k}\binom{k}{\ell}\binom{n-k-1}{k-\ell-1}\biggl(1-\frac{x^2}{\mu^2}\biggr)^\ell \end{align*} for $n\ge k\ge0$, where we used the identity $$ B_{n,k}\bigl(abx_1,ab^2x_2,\dotsc,ab^{n-k+1}x_{n-k+1}\bigr) =a^kb^n B_{n,k}(x_1,x_2,\dotsc,x_{n-k+1}) $$ and the formula $$ B_{n,k}(1!,2!,\dotsc,(n-k+1)!)=\binom{n-1}{k-1}\frac{n!}{k!} $$ for $n\ge k\ge0$. Further, by the Faa di Bruno formula \begin{equation*} \frac{\operatorname{d}^n[f\circ h(x)]}{\operatorname{d} x^n}=\sum_{k=0}^nf^{(k)}(h(x)) B_{n,k}\bigl(h'(x),h''(x),\dotsc,h^{(n-k+1)}(x)\bigr), \quad n\ge0, \end{equation*} we acquire \begin{align*} f^{(n)}(x)&=\sum_{k=0}^{n}\operatorname{e}^{h(x)} B_{n,k}\bigl(h'(x),h''(x),\dotsc,h^{(n-k+1)}(x)\bigr)\\ &=\operatorname{e}^{h(x)}\sum_{k=0}^{n}\frac{(-1)^{n+k}}{x^{n+k}}\frac{n!}{k!}\biggl(\frac{\lambda}{2}\biggr)^{k} \sum_{\ell=0}^{k}\binom{k}{\ell}\binom{n-k-1}{k-\ell-1}\biggl(1-\frac{x^2}{\mu^2}\biggr)^\ell\\ &=f(x)\frac{n!}{x^{2n}}\sum_{k=0}^{n}\frac{(-1)^{n-k}}{k!}\biggl(\frac{\lambda}{2}\biggr)^{k} x^{n-k} \sum_{\ell=0}^{k}\binom{k}{\ell}\binom{n-k-1}{k-\ell-1}\biggl(1-\frac{x^2}{\mu^2}\biggr)^\ell\\ &=\frac{f(x)}{2^n \mu^{2n} x^{2n}}\sum_{k=0}^{n}\lambda^k (-2x)^{n-k} \sum_{\ell=0}^{k}\binom{n}{\ell} \binom{n-k-1}{k-\ell-1} \frac{(n-\ell)!}{(k-\ell)!} \mu^{2(n-\ell)} \bigl(\mu^2-x^2\bigr)^\ell \end{align*} for $n\ge0$. In conclusion, we obtain the explicit derivative formula $$ \boxed{f^{(n)}(x)=\frac{f(x)}{2^n \mu^{2n} x^{2n}}\sum_{k=0}^{n}\lambda^k (-2x)^{n-k} \sum_{\ell=0}^{k}\binom{n}{\ell} \binom{n-k-1}{k-\ell-1} \frac{(n-\ell)!}{(k-\ell)!} \mu^{2(n-\ell)} \bigl(\mu^2-x^2\bigr)^\ell} $$ for $n\ge0$.

References

1. C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.
2. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.
3. https://qifeng618.wordpress.com/2018/03/19/some-papers-on-explicit-formulas-of-special-values-for-the-bell-polynomials-of-the-second-kind/.

Answer 5

We can give a fairly simple generating function for the polynomials $P_{\lambda,\mu}$ from which explicit formulas could be derived.

By Taylor's theorem, $$\sum_{n=0}^\infty f^{(n)}(x) \frac{y^n}{n!} = f(x+y).$$ Writing $P_{\lambda,\mu,n}$ for the proposer's $P_{\lambda,\mu}$, we then have \begin{align*} \sum_{n=0}P_{\lambda,\mu,n}(x) \frac{y^n}{n!} &= f(x+2\mu^2 x^2 y)/f(x)\\ &= \exp\left(\frac{-\lambda y (x^2-\mu^2+2\mu^2x^3y)}{1+2\mu^2xy}\right). \end{align*} So if we define polynomials $Q_n(a,b,c)$ by $$\sum_{n=0}^\infty Q_n(a,b,c)\frac{y^n}{n!} = \exp\left(\frac{ay+by^2}{1+cy}\right)$$ then $$P_{\lambda,\mu,n}(x) = Q_n\bigl(\lambda(\mu^2-x^2), -2\lambda\mu^2 x^3, 2\mu^2 x\bigr).$$

Answer 6

As I mentioned in my Math StackExhange answer Mathematica gives a difference root representation which leads to the formula

$$\frac{\partial^n e^{-\frac{\lambda\, (x-\mu)^2}{2\, \mu^2 x}}}{\partial x^n}=n!\, y(x,n)\tag{1}$$

where $y(x,n)$ is defined recursively as

$$y(x,n)=\left\{\begin{array}{cc} e^{-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}} & n=0 \\ e^{-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}} \frac{\lambda \left( \mu^2-x^2\right)}{2 \mu^2 x^2} & n=1 \\ e^{-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}} \frac{\lambda \left(\lambda \mu^4+\lambda x^4-2 \lambda \mu^2 x^2-4 \mu^4 x\right)}{8 \mu^4 x^4} & n=2 \\ -\frac{\left(-\lambda \mu^2+4 \mu^2 (n-3) x+\lambda x^2+8 \mu^2 x\right)\, y(x,n-1)+2 \left( \mu^2+\mu^2 (n-3)+\lambda x\right)\, y(x,n-2)+\lambda\, y(x,n-3)}{2 \mu^2 n x^2} & n\ge 3 \\ \end{array}\right.\tag{2}$$

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WolframAlpha gives the more general result

$$\frac{\partial^n e^{f(x)}}{\partial x^n}=e^{f(x)} \sum\limits_{k=0}^n \frac{1}{k!} \sum\limits_{j=0}^k (-1)^j \binom{k}{j} f(x)^j \frac{\partial ^nf(x)^{k-j}}{\partial x^n}\tag{3}$$

and for

$$f(x)=-\frac{\lambda\, (x-\mu)^2}{2\, \mu^2 x}\tag{4}$$

Mathematica simplifies formula (3) above to

$$\frac{\partial^n e^{-\frac{\lambda\, (x-\mu)^2}{2\, \mu^2 x}}}{\partial x^n}=e^{-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}} n! \sum\limits_{k=0}^n \frac{2^{-k}}{k!} \sum\limits_{j=0}^k (-1)^j \binom{k}{j} \left(-\frac{\lambda (x-\mu)^2}{\mu^2 x}\right)^j y_1(x,n)\tag{5}$$

where $y_1(x,n)$ is defined recursively as

$$y_1(x,n)=\left\{\begin{array}{cc} \left(-\frac{\lambda (x-\mu )^2}{\mu ^2 x}\right)^{k-j} & n=0 \\ \left(-\frac{\lambda (x-\mu )^2}{\mu ^2 x}\right)^{k-j-1} \frac{(\lambda (j-k) (x-\mu ) (\mu +x))}{\mu ^2 x^2} & n=1 \\ -\frac{(\mu (j-k-n+1)+x (j-k+2 (n-1)))\, y_1(x,n-1)+(j-k+n-2)\, y_1(x,n-2)}{n x (x-\mu )} & n\ge 2 \\ \end{array}\right.\tag{6}$$

which is a slightly simpler recursion than formula (2) above.

Answer 7

An alternative answer is \begin{equation*} \frac{\operatorname{d}^n}{\operatorname{d}x^n}\exp\biggl[-\frac{\lambda(x-\mu)^2}{2\mu^2x}\biggr]=\frac{f(x)}{2^n\mu^{2n}x^{2n}}\sum_{k=0}^{n}\binom{n}{k}(-\lambda)^{n-k} 2^{k}\mu^{2k}x^{2(n-k)} \sum_{\ell=0}^k \biggl(-\frac{\lambda}{2}\biggr)^\ell \Biggl[\sum _{j=\ell}^k s(k,j)(-1)^jS(j,\ell)\Biggr] x^{k-\ell} \end{equation*} for $n\ge0$, where $s(n,j)$ and $S(j,k)$ denote the Stirling numbers of the first and second kinds respectively.

References

1. S. Jin, B.-N. Guo, and F. Qi, Partial Bell polynomials, falling and rising factorials, Stirling numbers, and combinatorial identities, CMES Comput. Model. Eng. Sci. 132 (2022), no. 3, 781--799; available online at https://doi.org/10.32604/cmes.2022.019941.

Answer 8

In the references below, the derivative formula \begin{equation}\label{exp-frac1x-expans}\tag{QF1} \bigl(\operatorname{e}^{\pm1/t}\bigr)^{(n)} =(-1)^n{\operatorname{e}^{\pm1/t}}\sum_{k=0}^{n}(\pm1)^{k}\binom{n-1}{k-1}\frac{n!}{k!}\frac1{t^{n+k}}, \quad n\ge0 \end{equation} was derived alternatively. From \eqref{exp-frac1x-expans}, it follows that \begin{equation}\label{exp-frac1x-alpha}\tag{QF2} \bigl(\operatorname{e}^{\alpha/t}\bigr)^{(n)} =(-1)^n\operatorname{e}^{\alpha/t}\sum_{k=0}^{n}\alpha^{k}\binom{n-1}{k-1}\frac{n!}{k!}\frac1{t^{n+k}}, \quad n\ge0. \end{equation} Taking $\alpha=-\frac{\lambda}{2}$ in \eqref{exp-frac1x-alpha} gives \begin{equation*} \biggl[\exp\biggl(-\frac{\lambda}{2}\frac{1}{t}\biggr)\biggr]^{(k)} =(-1)^k\exp\biggl(-\frac{\lambda}{2}\frac{1}{t}\biggr) \sum_{\ell=0}^{k}\biggl(-\frac{\lambda}{2}\biggr)^{\ell} \binom{k-1}{\ell-1}\frac{k!}{\ell!}\frac1{t^{k+\ell}}, \quad k\ge0. \end{equation*} Therefore, in light of the Leibnitz rule for differentiation, we obtain \begin{gather*} \begin{aligned} f^{(n)}(x)&=\biggl[\exp\biggl(\frac{\lambda}{\mu}\biggr) \exp\biggl(-\frac{\lambda}{2\mu^2}x\biggr) \exp\biggl(-\frac{\lambda}{2}\frac{1}{x}\biggr)\biggr]^{(n)}\\ &=\exp\biggl(\frac{\lambda}{\mu}\biggr) \sum_{k=0}^{n}\binom{n}{k}\biggl[\exp\biggl(-\frac{\lambda}{2\mu^2}x\biggr)\biggr]^{(n-k)} \biggl[\exp\biggl(-\frac{\lambda}{2}\frac{1}{x}\biggr)\biggr]^{(k)} \end{aligned}\\ \begin{aligned} &=\exp\biggl(\frac{\lambda}{\mu}\biggr) \sum_{k=0}^{n}\binom{n}{k} \exp\biggl(-\frac{\lambda}{2\mu^2}x\biggr) \biggl(-\frac{\lambda}{2\mu^2}\biggr)^{n-k} (-1)^k\exp\biggl(-\frac{\lambda}{2}\frac{1}{x}\biggr) \sum_{\ell=0}^{k}\biggl(-\frac{\lambda}{2}\biggr)^{\ell} \binom{k-1}{\ell-1}\frac{k!}{\ell!}\frac1{x^{k+\ell}}\\ &=\exp\biggl(\frac{\lambda}{\mu}-\frac{\lambda}{2\mu^2}x-\frac{\lambda}{2}\frac{1}{x}\biggr) \sum_{k=0}^{n}\binom{n}{k} \biggl(-\frac{\lambda}{2\mu^2}\biggr)^{n-k} (-1)^k \sum_{\ell=0}^{k}\biggl(-\frac{\lambda}{2}\biggr)^{\ell} \binom{k-1}{\ell-1}\frac{k!}{\ell!}\frac1{x^{k+\ell}}\\ &=f(x)\frac{(-1)^nn!}{2^n\mu^{2n}x^{2n}} \sum_{k=0}^{n}\frac{\lambda^{n-k}}{(n-k)!} 2^{k}\mu^{2k} \sum_{\ell=0}^{k}\frac{(-\lambda)^{\ell}}{(2\ell)!!} \binom{k-1}{\ell-1}x^{2n-k-\ell}, \quad n\ge0. \end{aligned} \end{gather*} In conclusion, we obtain the derivative formula \begin{equation*} \boxed{f^{(n)}(x)=f(x)\frac{(-1)^nn!}{2^n\mu^{2n}x^{2n}} \sum_{k=0}^{n}\frac{\lambda^{n-k}}{(n-k)!} 2^{k}\mu^{2k} \sum_{\ell=0}^{k}\frac{(-\lambda)^{\ell}}{(2\ell)!!} \binom{k-1}{\ell-1}x^{2n-k-\ell}, \quad n\ge0.} \end{equation*}

References

1. S. Daboul, J. Mangaldan, M. Z. Spivey, and P. J. Taylor, The Lah numbers and the $n$th derivative of $e^{1/x}$, Math. Mag. 86 (2013), no. 1, 39--47; Available online at http://dx.doi.org/10.4169/math.mag.86.1.039.
2. F. Qi, An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers, Mediterr. J. Math. 13 (2016), no. 5, 2795--2800; available online at https://doi.org/10.1007/s00009-015-0655-7.
3. X.-J. Zhang, F. Qi, and W.-H. Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, Int. J. Open Probl. Comput. Sci. Math. 5 (2012), no. 3, 122--127; available online at https://doi.org/10.12816/0006128.
4. https://qifeng618.wordpress.com/2018/05/10/some-papers-related-to-the-function-e1-x-and-the-lah-numbers/