Lah number

Below is a table of values for the Lah numbers:

The row sums are $1,\; 1,\; 3,\; 13,\; 73,\; 501,\; 4051,\; 37633,\; \backslash dots$ {{OEIS|A000262}}.

Let $x^\{(n)\}$ represent the rising factorial $x(x+1)(x+2)\; \backslash cdots\; (x+n-1)$ and let $(x)\_n$ represent the falling factorial $x(x-1)(x-2)\; \backslash cdots\; (x-n+1)$. The Lah numbers are the coefficients that express each of these families of polynomials in terms of the other. Explicitly,$$x^\{(n)\}\; =\; \backslash sum\_\{k=0\}^n\; L(n,k)\; (x)\_k$$and$$(x)\_n\; =\; \backslash sum\_\{k=0\}^n\; (-1)^\{n-k\}\; L(n,k)x^\{(k)\}.$$For example,$$x(x+1)(x+2)\; =\; \{\backslash color\{red\}6\}x\; +\; \{\backslash color\{red\}6\}x(x-1)\; +\; \{\backslash color\{red\}1\}x(x-1)(x-2)$$and$$x(x-1)(x-2)\; =\; \{\backslash color\{red\}6\}x\; -\; \{\backslash color\{red\}6\}x(x+1)\; +\; \{\backslash color\{red\}1\}x(x+1)(x+2),$$

where the coefficients 6, 6, and 1 are exactly the Lah numbers $L(3,\; 1)$, $L(3,\; 2)$, and $L(3,\; 3)$.

The Lah numbers satisfy a variety of identities and relations.

In Karamata–Knuth notation for Stirling numbers$$L(n,k)\; =\; \backslash sum\_\{j=k\}^n\; \backslash left[\{n\backslash atop\; j\}\backslash right]\; \backslash left\backslash \{\{j\backslash atop\; k\}\backslash right\backslash \}$$where $\backslash left[\{n\backslash atop\; j\}\backslash right]$ are the unsigned Stirling numbers of the first kind and $\backslash left\backslash \{\{j\backslash atop\; k\}\backslash right\backslash \}$ are the Stirling numbers of the second kind.

:$L(n,k)\; =\; \{n-1\; \backslash choose\; k-1\}\; \backslash frac\{n!\}\{k!\}\; =\; \{n\; \backslash choose\; k\}\; \backslash frac\{(n-1)!\}\{(k-1)!\}\; =\; \{n\; \backslash choose\; k\}\; \{n-1\; \backslash choose\; k-1\}\; (n-k)!$

:$L(n,k)\; =\; \backslash frac\{n!(n-1)!\}\{k!(k-1)!\}\backslash cdot\backslash frac\{1\}\{(n-k)!\}\; =\; \backslash left\; (\backslash frac\{n!\}\{k!\}\; \backslash right\; )^2\backslash frac\{k\}\{n(n-k)!\}$

:$k(k+1)\; L(n,k+1)\; =\; (n-k)\; L(n,k)$, for $k>0$.

The Lah numbers satisfy the recurrence relations$$$$

\begin{align} L(n+1,k) &= (n+k) L(n,k) + L(n,k-1) \\

&= k(k+1) L(n, k+1) + 2k L(n, k) + L(n, k-1)

\end{align}

where $L(n,0)=\backslash delta\_n$, the Kronecker delta, and $L(n,k)=0$ for all $k\; >\; n$.

:$\backslash sum\_\{n\backslash geq\; k\}\; L(n,k)\backslash frac\{x^n\}\{n!\}\; =\; \backslash frac\{1\}\{k!\}\backslash left(\; \backslash frac\{x\}\{1-x\}\; \backslash right)^k$

The n-th derivative of the function $e^\backslash frac1\{x\}$ can be expressed with the Lah numbers, as follows{{cite journal |last1=Daboul |first1=Siad |last2=Mangaldan |first2=Jan |last3=Spivey |first3=Michael Z. |last4=Taylor |first4=Peter J. |year=2013 |title=The Lah Numbers and the nth Derivative of $e^\{1\backslash over\; x\}$ |journal=Mathematics Magazine |volume=86 |pages=39–47 |doi=10.4169/math.mag.86.1.039 |jstor=10.4169/math.mag.86.1.039 |s2cid=123113404 |number=1}}$$\backslash frac\{\backslash textrm\; d^n\}\{\backslash textrm\; dx^n\}\; e^\backslash frac1x\; =\; (-1)^n\; \backslash sum\_\{k=1\}^n\; \backslash frac\{L(n,k)\}\{x^\{n+k\}\}\; \backslash cdot\; e^\backslash frac1x.$$For example,

$\backslash frac\{\backslash textrm\; d\}\{\backslash textrm\; dx\}\; e^\backslash frac1x\; =\; -\; \backslash frac\{1\}\{x^2\}\; \backslash cdot\; e^\{\backslash frac1x\}$

$\backslash frac\{\backslash textrm\; d^2\}\{\backslash textrm\; dx^2\}e^\backslash frac1\{x\}\; =\; \backslash frac\{\backslash textrm\; d\}\{\backslash textrm\; dx\}\; \backslash left(-\backslash frac1\{x^2\}\; e^\{\backslash frac1x\}\; \backslash right)=\; -\backslash frac\{-2\}\{x^3\}\; \backslash cdot\; e^\{\backslash frac1x\}\; -\; \backslash frac1\{x^2\}\; \backslash cdot\; \backslash frac\{-1\}\{x^2\}\; \backslash cdot\; e^\{\backslash frac1x\}=\; \backslash left(\backslash frac2\{x^3\}\; +\; \backslash frac1\{x^4\}\backslash right)\; \backslash cdot\; e^\{\backslash frac1x\}$

$\backslash frac\{\backslash textrm\; d^3\}\{\backslash textrm\; dx^3\}\; e^\backslash frac1\{x\}\; =\; \backslash frac\{\backslash textrm\; d\}\{\backslash textrm\; dx\}\; \backslash left(\; \backslash left(\backslash frac2\{x^3\}\; +\; \backslash frac1\{x^4\}\backslash right)\; \backslash cdot\; e^\{\backslash frac1x\}\; \backslash right)\; =\; \backslash left(\backslash frac\{-6\}\{x^4\}\; +\; \backslash frac\{-4\}\{x^5\}\backslash right)\; \backslash cdot\; e^\{\backslash frac1x\}\; +\; \backslash left(\backslash frac2\{x^3\}\; +\; \backslash frac1\{x^4\}\backslash right)\; \backslash cdot\; \backslash frac\{-1\}\{x^2\}\; \backslash cdot\; e^\{\backslash frac1x\}\; =-\backslash left(\backslash frac6\{x^4\}\; +\; \backslash frac6\{x^5\}\; +\; \backslash frac1\{x^6\}\backslash right)\; \backslash cdot\; e^\{\backslash frac\{1\}\{x\}\}$

Generalized Laguerre polynomials $L^\{(\backslash alpha)\}\_n(x)$ are linked to Lah numbers upon setting $\backslash alpha\; =\; -1$$$n!\; L\_n^\{(-1)\}(x)\; =\backslash sum\_\{k=0\}^n\; L(n,k)\; (-x)^k$$This formula is the default Laguerre polynomial in Umbral calculus convention.{{Cite journal |last1=Rota |first1=Gian-Carlo |last2=Kahaner |first2=D |last3=Odlyzko |first3=A |date=1973-06-01 |title=On the foundations of combinatorial theory. VIII. Finite operator calculus |journal=Journal of Mathematical Analysis and Applications |language=en |volume=42 |issue=3 |pages=684–760 |doi=10.1016/0022-247X(73)90172-8 |issn=0022-247X|doi-access=free }}

In recent years, Lah numbers have been used in steganography for hiding data in images. Compared to alternatives such as DCT, DFT and DWT, it has lower complexity of calculation—$O(n\; \backslash log\; n)$—of their integer coefficients.{{Cite journal |last1=Ghosal |first1=Sudipta Kr |last2=Mukhopadhyay |first2=Souradeep |last3=Hossain |first3=Sabbir |last4=Sarkar |first4=Ram |year=2020 |title=Application of Lah Transform for Security and Privacy of Data through Information Hiding in Telecommunication |journal=Transactions on Emerging Telecommunications Technologies |volume=32 |issue=2 |doi=10.1002/ett.3984|s2cid=225866797 }}{{Cite web |title=Image Steganography-using-Lah-Transform |url=https://in.mathworks.com/matlabcentral/fileexchange/78751-image-steganography-using-lah-transform |website=MathWorks|date=5 June 2020 }}

The Lah and Laguerre transforms naturally arise in the perturbative description of the chromatic dispersion.{{Cite journal|last1=Popmintchev|first1=Dimitar|last2=Wang|first2=Siyang|last3=Xiaoshi |first3=Zhang |last4=Stoev|first4=Ventzislav|last5=Popmintchev|first5=Tenio|date=2022-10-24 |title=Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion|journal=Optics Express|volume=30|issue=22|pages=40779–40808|doi=10.1364/OE.457139|doi-access=free|pmid=36299007 |bibcode=2022OExpr..3040779P}}{{cite arXiv|last1=Popmintchev|first1=Dimitar|last2=Wang|first2=Siyang|last3=Xiaoshi|first3=Zhang |last4=Stoev|first4=Ventzislav|last5=Popmintchev|first5=Tenio|date=2020-08-30|title= Theory of the Chromatic Dispersion, Revisited |class=physics.optics |eprint=2011.00066}}

In Lah-Laguerre optics, such an approach tremendously speeds up optimization problems.

• Stirling numbers
• Pascal matrix

• The signed and unsigned Lah numbers are respectively {{OEIS|A008297}} and {{OEIS|A105278}}

{{Classes of natural numbers}}

{{DEFAULTSORT:Lah Number}}

Category:Factorial and binomial topics

Category:Integer sequences

Category:Triangles of numbers