Shadows of Simplicity

The Pincherle Derivative and the Appell Raising Operator

Posted on June 12, 2016 by Tom Copeland

The raising and lowering operators $R$ and $L$ for a sequence of functions $\psi_n(x)$ , with $n= 0,1, 2, ...$ and $\psi_0(x)=1$ , defined by

$R \; \psi_n(x) = \psi_{n+1}(x)$ and $L \; \psi_n(x) = n \; \psi_{n-1}(x)$

have the commutator relation

$[L,R] = LR-RL = 1$

with respect to action on the space spanned by this sequence of functions.

If for $m$ any particular natural number

$[L^m,R] = m \; L^{m-1} = \frac{d}{dL}L^m$ ,

then

$mL^m =L \; [L^m,R] = L^{m+1}R - LRL^{m}$

$= L^{m+1}R - (1+RL)L^{m} = L^{m+1}R - RL^{m+1} - L^m$ ,

implying

$[L^{m+1},R] = (m+1) L^{m} = \frac{d}{dL}L^{m+1}$ .

Since this holds for $m=1$ , the relation holds for all natural numbers, and formally for a function $f(x)=e^{a.x}$ analytic about the origin (or a formal power series or exponential generating function)

$[f(L),R] = [e^{a.L},R]= \frac{d}{dL}e^{a.L} = a. \; e^{a.L}=\frac{d}{dL}f(L)$ .

The reader should be able to modify the argument to show that also

$[L,f (R)] = \frac {d}{dR}f(R)$ .

An important application of the Pincherle derivative is to connecting different reps of the raising operators of Appell sequences:

The iconic ladder operators are $L_p = D = d/dx$ and $R_p=x$ for the the powers $x^n$ , the prototypical Appell sequence of polynomials (see the post Bernoulli Appells for more on Appell sequences), so

$[f(D),x] = [f(L_p),R_p]=\frac {d}{dD}f(D) = f'(D)$ ,

and

$f(D)^{-1} \; x \; f(D) =x-x+f(D)^{-1} \; x \; f(D)$

$= x - f(D)^{-1} \; [f(D,x] = x - f(D)^{-1}f'(D) = x-\frac {d}{dD} \; \ln(f(D))$

$= \frac{d}{dt}|_{t=D} \; \ln[e^{x \; t} / f(t)] = \frac{d}{dt}|_{t=D} \; \ln[e^{x \; t} e^{c.t}]=\frac{d}{dt}|_{t=D} \; \ln[e^{(c.+ x) \; t}]$ .

If $f(0) = 1$ , then $R_A = f(D)^{-1} \; x \; f(D)= x- \frac{d}{dD} \; \ln(f(D))$ is the raising operator (see Bernoulli Appells) for an Appell sequence $p_n(x)$ with moments given by the coefficients $(c.)^n = c_n$ of the Taylor series for $f(x)^{-1}$ , i.e., $(c. +x)^n = p_n(x)$ ; lowering operator $L_A = D$ ; and e.g.f. $e^{tp.(x)} = e^{x \; t} / f(t)= e^{(c. + x)t}$ .

Related stuff:

“The many avatars of a simple algebra” by Coutinho

http://mathoverflow.net/questions/97512/in-splendid-isolation/98213#98213

https://tcjpn.wordpress.com/2014/08/03/goin-with-the-flow-logarithm-of-the-derivative/

https://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/

https://tcjpn.wordpress.com/2015/11/21/the-creation-raising-operators-for-appell-sequences/

https://tcjpn.wordpress.com/2014/12/10/appells-for-the-bernoullis/

http://oeis.org/A263634

Posted in Math | Tagged Appell sequences, Commutator, Ladder operators, Lowering and raising operators, Pincherle derivative | Leave a comment

Generators, Inversion, and Matrix, Binomial, and Integral Transforms

Posted on December 21, 2015 by Tom Copeland

Generators, Inversion, and Matrix, Binomial, and Integral Transforms is a belated set of notes (pdf) on a derivation of a generating function for the row polynomials of OEIS-A111999 from its relation to the compositional inversion (a Lagrange inversion formula, LIF) presented in A133932 of invertible functions represented umbrally as logarithmic series $-ln(1-b.x)$ . The results show that A111999 is a natural reduction of A133932.

Along the way, more general results are given involving the relations among Borel-Laplace transforms; compositional inversion in general; binomial transforms of rows, columns, and diagonals of matrices; infinitesimal generators; and the generating functions and reversals of binomial and Appell Sheffer polynomials, in particular the cycle index polynomials of the symmetric groups, or partition polynomials of the refined Stirling numbers of the first kind A036039.

A table has been added in Appendix III to illustrate how the analysis applies to the two other complementary LIFs A134685 , based on the refined Stirling numbers of the second kind A036040 (a refinement of the Bell / Touchard / exponential polynomials A008277), and A133437, based on the refined Lah numbers A130561 (a refinement of the Lah polynomials A008297, A105278, normalized Laguerre polynomials of order -1).

Posted in Math, Uncategorized | Tagged Appell polynomial sequences, Bell polynimials, Binomial Sheffer sequences, Borel-Laplace transform, Compositional inverse, Compositional inversion, Cycle index polynomials of symmetric group, Differential operators, Euler binomial transforms, Exponential generating functions, Infinitesimal generators, Lagrange inversion, Lah polynomials, OEIS-133437, OEIS-134685, OEIS-A036039, OEIS-A036040, OEIS-A111999, OEIS-A130561, OEIS-A133932, Ordinary generating function, Reverse polynomials, Series reversion, Sheffer polynomials, Stirling numbers of the first kind, Stirling numbers of the second kind, Upper incomplete gamma function | Leave a comment

The Creation / Raising Operators for Appell Sequences

Posted on November 21, 2015 by Tom Copeland

The Creation / Raising Operators for Appell Sequences is a pdf presenting reps of the raising operator $R$ and its exponentiation $\exp(tR)$ for normal and logarithmic Appell sequences of polynomials as differential and integral operators. The Riemann zeta and digamma, or Psi, function are connected to fractional calculus and associated Appell sequences for a characteristic genus discussed by Libgober and Lu.

Additional notes:

Using the inverse Mellin transform rep of the Dirac delta function given in an earlier entry leads to the integral kernel $K(x,-m) = H(1-x) \frac{(1-x)^{-m-1}}{(-m-1)!}=(-1)^m \frac{d}{dx}^m \delta(1-x)=\delta^{(m)}(1-x)$ for $K(x,t)$ on page $10$ .

(Added 9/8/2016) The post Bernoulli Appells contains yet another rep for an Appell polynomial raising operator:

$R = e^{B.(0)D_x} \; x \; e^{\hat{B}(0)D_x} = x - x + e^{B.(0)D_x} \; x \; e^{\hat{B}(0)D_x} = x - e^{B.(0)D_x}[e^{\hat{B}(0)D_x},x]$ ,

which holds for any sequence of Appell polynomials $B_n(x)$ and its umbral inverse Appell sequence $\hat{B}(x)$ . See OEIS A263634 for matrix reps of the raising op.

(Added 9/15/2016) For the convolution rep of the derivative op and its relation to the Mellin transform in Part III of the pdf, see the post Note on the Inverse Mellin Transform and the Dirac Delta Function on the inverse Mellin transform rep for the derivative of the Dirac delta :

$\displaystyle \delta^{'}(y-x) = \frac{d}{dy} \delta(y-x) = \frac{d}{dy} \int_{\sigma-i \infty }^{\sigma + \infty} y^{s-1}x^{-s}ds =\int_{\sigma-i \infty }^{\sigma + \infty} (s-1) y^{s-2}x^{-s}ds$ ,

so in this sense

$\displaystyle \delta^{'}(1-x) = \int_{\sigma-i \infty }^{\sigma + \infty} (s-1) x^{-s}ds = H(1-x) \frac{(1-x)^{t-1}}{(t-1)!}|_{t=-1}$ .

Errata:

Pg. 2: $A(\phi.(:xD_x:+t)$ should be $A(\phi.(:xD_x:)+t)$ .

Pg. 9: $e^{tR_x}x^s$ should be $e^{tR_x}$ .

Related Stuff:

“Two applications of elementary knot theory to Lie algebras and Vassiliev invariants” by Bar-Natan, Le, Thurston (note formulas containing sinh). See also the Thurston paper referenced in the post Bernoulli Appells.

Posted in Math | Tagged Appell sequences, Convolution integrals, Convolution operators, Creation and annihilation operators, Cumulant generating function, Cycle index polynomials, Differential operators, Digamma function, Formal cumulants and moments, Fractional calculus, Infiinitesimal generators, Infinigens, Infinitesimal generators, Inverse Mellin transform, Ladder operators, Laguerre operator, Logarithm of the derivative operator, Mellin transform, Mellin transform interpolation of operators, Operator calculus, Psi function, Raising and lowering operators, Riemannn zeta function, Stirling polynomials of the first and second kinds, Symmetric group | 1 Comment

The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera

Posted on October 12, 2015 by Tom Copeland

The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera

(This site was not correctly updating, so the notes were transcribed to this pdf.)

Addendum to The Elliptic Lie Triad

Continue reading →

Posted in Math | Tagged Bernoulli numbers, Chebyshev polynomials, Combinatorics, Compositional inverse, Differential operators, Elliptic curves, Elliptic formal group law, Elliptic genera, Enumerative combinatorics, Euler numbers, Eulerian numbers, Evolution equations, Faber polynomials, Geodesics of Virasoro-Bott group, Grassmann polynomials, Grassmannian, Hyperbolic tangent, Infinigens, Infinitesimal generators, KdV, Korteweg-de Vries equation, L-genus, Lie algebra, Lucas (Cardan) polynomials, Refined Eulerian numbers, Riccati, Schwarzian derivative, SL2, Solitons, Swiss knife polynomials, Viscous Burgers equation | Leave a comment

The Kervaire-Milnor Formula

Posted on October 4, 2015 by Tom Copeland

The K-M formula and its ingredients are presented in

1) Bernoulli numbers and the unity of mathematics by Barry Mazur, p.14, Secs. 4, 5, and 6

2) Differential topology forty-six years later by Milnor

3) Homotopy group of spheres Wikipedia

4) Exotic sphere Wikipedia

5) J-homomorphism Wikipedia

6) A history of the Arf-Kervaire invariant problem by Snaith

7) Bernoulli numbers, homotopy groups, and a theorem of Rohlin by Milnor and Kervaire

The K-M formula, as presented by Mazur, is

$card [\Theta_{4k-1}] = R(k) \; card[ H_{4k-1}] \; B_{2k}/2k$

where (if I interpret Mazur, and Milnor, correctly) $\Theta_{j}$ is the group of homotopy spheres up to h-cobordism, or essentially the set of all oriented diffeomorphism classes of closed smooth homotopy $n$ -spheres; $R(k)=2^{2k-2}(2^{2k-1}-1)$ for odd $k$ and twice that for even $k$ ; $H_{j}$ is the group of stable homotopy classes of continuous maps from the $(m+j)$ -sphere to the $j$ -sphere, and $B_n$ are the Bernoulli numbers.

Continue reading →

Posted in Math | Tagged Alternating permutations, Appell sequences, Bernoulli numbers, cycle graph, Dirichlet eta function, Euler numbers, Exotic sphere, exponential generating function, Fermi-Dirac distribution, homotopy group, hypersphere, integer sequences, Kervaire Milnor formula, n-sphere, parallelizable manifold, Riemann zeta function | 2 Comments

Snakes in the Appell Orchard

Posted on October 1, 2015 by Tom Copeland

The Euler-Bernoulli numbers: what they count and associations to algebraic geometry, elliptic curves, and differential ops. Coming soon.

Posted in Uncategorized | Leave a comment

Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops

Posted on September 16, 2015 by Tom Copeland

Interpolations of the derivative operator $D_x^n \; ,$ the fundamental ordered op $:xD_x:^n=x^nD_x^n \; ,$ the Laguerre op $:D_xx:^n = D^nx^n \; ,$ the shifted Laguerre op $(xD_xx)^n = x^nD_x^nx^n \; ,$ and the generalized Scherk-Witt Lie ops $(x^{1+y}D_x)^n$ to the fractional operators $D_x^s\; , \; :xD_x:^s = x^sD_x^s \; , \; :D_xx:^s = D_x^sx^s \; , \; (xD_xx)^s = x^sD_x^sx^s \; , \;$ and $(x^{1+y}D_x)^s$ are consistently achieved using the Mellin transform of the negated e.g.f.s of the differential ops. Associated infinitesimal generators (infinigens) are then determined for each fractional op and related to the raising ops for associated Appell sequences.

Continue reading →

Posted in Math | Tagged Associated Laguerre polynomials, Compositional inverse, Differential operator interpolation, exponential generating function, Falling factorials, Infinigens, Infinitesimal generators, Integral curves, Inverse Mellin transform, Lie Witt algebra, Mellin transform interpolation, Rising factorials, Vector fields | Leave a comment

	The Pincherle Deriva… on Bernoulli Appells
	The Pincherle Deriva… on The Creation / Raising Operato…
	A Note on the Pinche… on Infinigens, the Pascal Triangl…
	Tom Copeland on The Kervaire-Milnor Formula
	A Note on the Pinche… on Goin’ with the Flow: Log…

Shadows of Simplicity

The Pincherle Derivative and the Appell Raising Operator

Generators, Inversion, and Matrix, Binomial, and Integral Transforms

The Creation / Raising Operators for Appell Sequences

The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera

The Kervaire-Milnor Formula

Snakes in the Appell Orchard

Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops

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