Witt Differential Generator for Special Jack Symmetric Functions / Polynomials

Exploring some relations among the multinomial coefficients of OEIS A036038 and the compositional inversion formulas of A134264, A248120, and A248927, related to numerous combinatorial structures and areas of analysis, including noncrossing partitions and free probability,  I came across the Jack symmetric functions J_n^{\alpha}(x_1,x_2, ...) in an infinite number of variables as presented in “Some combinatorial properties of the Jack symmetric functions” by Stanley.

Equation a) on page 80 of his paper leads to an umbral generating formula for the related Jack symmetric polynomials (JSP)

 J_n^{\alpha}(x_1,x_2, ...,x_{n+1}) = (\hat{x}_1 + \hat{x}_2 + \cdots + \hat{x}_{n+1})^n,

where \hat{x}_k is to be regarded as a regular variable until the expression is reduced to monomials at which time it is to be evaluated as  \hat{x}_k^j = s_j (\alpha) \; x_k^j with s_j(\alpha)  =  1 (1+\alpha)(1+2 \alpha) \cdots (1+(j-1) \alpha), essentially the row polynomials of A094638 comprised of the Stirling numbers of the first kind.  For example,

J_2^{\alpha}(x_1,x_2,x_3) = (\hat{x}_1 + \hat{x}_2  + \hat{x}_3)^2

= \sum_{k=1}^3 \; \hat{x}_k^2 \; + \; 2 \; \sum_{i,j=1 ; i \neq j}^3 \; \hat{x}_i \; \hat{x}_j = m_{[2]}(\hat{x}_1,\hat{x}_2,\hat{x}_3) \; + \; 2 \; m_{[1,1]}(\hat{x}_1,\hat{x}_2,\hat{x}_3)

= s_2(\alpha) \; m_{[2]}(x_1,x_2,x_3) \; + \; 2 \; s_1(\alpha) \; s_1(\alpha) \; m_{[1,1]}(x_1,x_2,x_3)

= (1+\alpha) \; m_{[2]}(x_1,x_2,x_3) \; + \; 2 \; m_{[1,1]}(x_1,x_2,x_3) ,

where the polynomial has been expressed in the symmetric monomial polynomials (SMP), easily extended to an indefinite number of variables. The factors multplying the SMPs are the multinomial coefficients of A036038 and remain independent of the number of variables. Each summand of an SMP has the same configuration of exponents and subscripts, allowing the products of s_k (\alpha) to be easily determined and factored out after the umbral evaluation.

MOPS: Multivariate orthogonal polynomials (symbolically)” by Dumitriu, Edelman, and Shuman contain examples for the third and fourth JSPs, but the third has the coefficients erroneously transposed for the factorial.

Using the operator identities in A094638, a Rodriques-like generator for the JSPs can be devised.

(z^{1+y} \; D_z)^n = z^{ny} \; zD_z \; s_n (y \; zD_z),


z^{-1} \; z^{-ny} (z^{1+y} \; D_z)^n \; z = z^{-1} \;  zD_z \;s_n(y \; zD_z) \; z = s_n (y),

and, with w_k = 1/z_k^y

(\hat{x}_k)^j = z_k^{-1} \; w_k^j \; (z_k^{1+y} \; D_{z_k})^j \; x_k^j \; z_k = s_j(y) \; x_k^j

= z_k^{-1} (x_k \; \hat{w}_k \; z_k^{1+y} \; D_{z_k})^j \; z_k ,

with \hat{w}_k treated as independent of z_k with respect to the derivations and, after being passed unscathed to the left of all derivations, is finally evaluated as   \hat{w_k} = 1/z_k^y .

Then, with Q = \prod_{k>0} \; z_k

J_n^y = Q^{-1} \; [\sum_{k>0} \; x_k \; \hat{w}_k \; z_k^{1+y} \; D_{z_k}]^n \; Q .

This may be generalized just as the Laguerre polynomials are to the associated Laguerre polynomials by conjugating with Q^u rather than Q .

A generating function for the full set of JSPs is

\exp[t\; J.^y] = Q^{-1} \; \prod_{k>0} \; \exp[t \; x_k \; \hat{w_k} \; z_k^{1+y} \; D_{z_k}] \; Q .

This can be evaluated by noting, with

\sigma = z^{-y}/(-y) and L_y = z^{1+y} \; \frac{d}{dz} = \frac{d}{d\sigma},


e^{\beta \; L_y} f (z) = e^{\beta \; \frac{d}{d\sigma}} \; f[(-y\sigma)^{-1/y}] = f[(-y (\sigma +\beta))^{-1/y}].

Then, for \beta =  t \; x_k \; \hat{w_k} and f(z)= z , the operation reduces to

[-y(\sigma+\beta)]^{-1/y} = [z^{-y}-t \; y \; x_k \; \hat{w_k}]^{-1/y} = z / [1-t \;y \; x_k \; \hat{w}_k \; z^y]^{1/y}.

Then the generator gives, since ultimately \hat{w_k} = 1/z_k^y ,

e^{t \; J.^y} = \prod_{k>0} \;  [1-t \; y \; x_k]^{-1/y},


e^{t \; J.^\alpha} = \prod_{k>0} \;  [1-t \; \alpha \; x_k]^{-1/\alpha}.

You can use the generalized Leibnitz formula to relate this back to the multinomial coefficients:

D_t^m \; \prod_{k=1}^{n+1} g_k(t) = (D_1  + D_2 + \cdots + D_{n+1})^m \; \prod_{k} g_k(t)

where D_j acts as d/dt only on g_j(t).

The e.g.f. is naturally consistent with the e.g.f. for A094638, which with a change of notation is

e^{s.(\alpha)x} = (1-\alpha x)^{-1/ \alpha},

and is an e.g.f. for plane m-ary trees with \alpha = m-1, so

\prod_k \; (1-\alpha x_kt)^{-1/ \alpha} = \prod_k \; e^{s.(\alpha)x_kt} = \prod_k \; e^{\hat{x}_k t} = e^{tJ.^{\alpha}}.

Therefore, the discussions in A134264 and the Hirzebruch criterion post below on repeated exponentiaton of an e.g.f., binomial convolutions, and umbral substitution apply to these calculations when x_k=x , giving connections among the OEIS entries cited at the top here and the multinomial coefficients.

Taking the log of the e.g.f. gives a relation between the symetric power sum polynomials / functions of the variables / indeterminates and the cumulants formed from the JSPs through A127671, or A263634.

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An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

(This is a duplicate of a Mathoverflow question posed in Oct 2014 that ran the gauntlet of the OCD cadre there–the demonstrative ones I assume avoid stepping on cracks in the sidewalk and become obstructive, hostile, and/or jealous at popular questions they can’t immediately circumscibe within their own rigid, limited views on how and what math should be done, naturally incuding terminology: Bernoullians!? Blasphemy! See also this blog post. In a couple of days the question got around 800 views, 9 upvotes, 7 downvotes, two or three close requests, several unconstructive/petty comments on rewriting/formatting, but no answers. Recognizing, from experience on MO, the futility of further refining the question and having pursued a plausible partial answer myself, I deleted most of it. Here is the full text so that I and others can easily follow related references.)

Prelude on Two Threads

Last month (Sept. 2014) the workshop New Geometric Structures in Scattering Amplitudes was hosted by CMI with the following partial overview:

Recently, remarkable mathematical structures have emerged in the study of scattering amplitudes, revealing deep links to algebraic geometry, arithmetic and combinatorics. There have been many exciting dual representations of amplitudes including

  •  polytopes, twistor diagrams and the positive grassmannian leading to the concept of the amplituhedron,
  •  polylogs, multizeta values and multiloop integrals,
  •  …

Each of these disparate ideas and methods have their own remarkable successes, and indeed have led to important progress in related areas of mathematics. They also face obstacles that they need to overcome in order to become important tools in resolving the most important open conjectures in the field and for the explicit construction of amplitudes.

Classic Number Clans in the Tapestry

It’s a rich tapestry in the making (note this Dec. workshop), weaving together much interesting math and physics, in which I would like to follow some threads particularly intriguing to me (since they relate to different perspectives on the combinatorics of polytopes and simplicial complexes related to Lagrange compositional inversion).

The two bullets above relate to two threads

I ) T_V: Totally non-negative Grassmannians G^+(k,n) ~ matroid polytopes P_M ~ Vol(P_M) ~ degrees of toric varieties ~ number of solutions of polynomial equations related to scattering amplitudes in twistor string theory from volume/contour integrals over G^+.

II) T_C: G^+ ~ Stasheff associahedra ~ cluster algebras/coordinates ~ generalized polylogarithms ~ MZVs ~ multiloop integrals for scattering amplitudes

It was Marni Sheppeard through her paper “Constructive Motives and Scattering]” who first gave me a docent’s tour of this tapestry. In particular, she points out some threads interweaving Grassmannians, associahedra, cluster algebra, generalized permutohedra, volumes of hypersimplices and the Eulerian numbers, volumes (and binary trees) enumerated by the Narayana numbers, and scattering amplitudes, among others.

And, Lauren Williams in “Enumeration of totally positive Grassmann cells]” develops a polynomial generating function A_{k,n}(q) whose q^d coefficient is the number of totally positive cells in G^+(k,n) that have dimension d and goes on to show that for the binomial transform \hat{E}_{k,n}(q)=q^{k-n}\sum^n_{i=0} (-1)^i \binom{n}{i} A_{k,n-i}(q) that \hat{E}_{k,n(}(1)=E_{k,n}, the Eulerians, and \hat{E}_{k,n}(0)=N_{k,n}, the Narayanaians. She reiterates this in her presentation “The Positive Grassmannian (a mathematician’s perspective)” and relates G+ to soliton shallow-water-wave solutions of a KP equation, noting the roles of G^+ in computing scattering amplitudes in string theory, a relation to free probability, and the occurrence of the Eulerians and Narayanaians in the BCFW recurrence and twistor string theory.

The number clans that appear in the tapestry are listed below along with some associations. (The Wardians seem peripheral for the moment, but they do lead to the associahedra through fans and phylogenetic trees, and the refined ones can be scaled to the refined f-vectors of the associahedra through their relation to Lagrange inversion.)

Some relations to number clans:

Eulerians, E_{n,k} (A008292, refined-A145271):

h-vectors of simplicial complexes dual to the permutohedra ~ volumes of hypersimplices ~ degrees of varieties ~ number of solutions of polynomials for scattering amplitudes

Catalanians, C_{n,k} (A033282, refined-A133437):

f-vectors of Stasheff associahedra (for Coxeter group A_n), related to dissections of polygons (with the Catalans, # of vertices, enumerating the triangulations) ~ structure of associahedra reflects cluster algebra relations

Narayanaians, N_{n,k} (A001263, refined-A134264):

h-vectors of the simplicial complex dual to the Stasheff associahedra, sum to the Catalan numbers, enumerate non-crossing partitions on [n] and refinement of binary trees (right-pointing leaves), refined Narayanaians relate number of connected positroids on [n] to the total number of positroids through an inversion (see A134264)

Wardians, W{n,k} (A134991, refined-A134685):

f-vectors of the Whitehouse simplicial complex associated with the tropical Grassmannians G(2,k) and phylogenetic trees (Bergman matroids?), related to enumeration of partitions of 2n objects.


A) Are there other perspectives on this tapestry involving these classic number clans, i.e., other ways in which these clans show up in the tapestry?

B) Can someone give a more cogent overview of these two threads and their relation to the classic number arrays?


(More details for the interested.)

Thread T_V:

1) Alcoved polytopes I, Lam and Postnikov, pages 1, 2, 18, and 21,

keywords: volumes, hypersimplices, Eulerian, grassmannian manifold, torus orbit

2) Matroid polytopes and volumes, Ardila, Benedetti, and Doker, page 6,

keywords: generalized permutohedra, grassmannian, torus orbit, volumes

3) Loops, Legs and Twistors, Spradlin,

keywords: contour integral, amplitudes, polynomials equations, Eulerian numbers

4) A note on polytopes for scattering amplitudes, Arkani-Hamed, Bourjaily, Cachazo, Hodges, and Trnka, pages 5-10,

keywords: twistor theory, volumes, areas, contour integral, differential forms

5) Scattering in three dimensions from rational maps, Cachazo, He, and Yuan, pages 4 and 11,

keywords: Eulerian numbers, scattering equations

6) Scattering Equations, Yuan

keywords: vanishing, quadratic differential, polynomoal maps, Eulerian numbers

7) Gravity in Twistor Space and its Grassmannian Formulation, Cachazo, Mason, and Skinner, pages 18 and 19,

keywords: Eulerian number, marked points

Thread T_C;

8) Matching polytopes, toric geometry, and the non-negative part of the Grassmannian, Postnikov, Speyer, and Williams, pages 1-2 and 12-13,

keywords: Grassmannians, matroid polytope, toric variety, cluster algebra

9) Cluster Polylogarithms for Scattering Amplitudes, Golden, Paulos, Spradlin, and Volovich, pages 8-13,

keywords: Stasheff associahedra, cluster functions

10) Studying Quantum Field Theory, Todorov, pages 17-20,

keywords: Catalan numbers, iterated integrals, simplices, polylogarithms

Additional notes on number clans, combinatorics, polytopes, and algebraic geometry:

11) On the Hilbert polynomials and Hilbert series of homogeneous projective varieties, Gross and Wallach, pages 13 and 14,

keywords: Grassmannian, Catalan numbers, Narayana numbers

12) Eulerian polynomials, Hirzebruch

13) New moduli spaces of pointed curves and pencils of flat connections, Losev and Manin, page 8, (Eulerians misnamed as the the Euler numbers)

14) For the Eulerian polynomial, see my recent entry in the formula section of A008292 and the associated links to Lenart and Zanoulline, and Buchstaber and Bunkova.

15) For the Narayana polynomials, see my recent example in A134264 and the associated reference to Ardila, Rincon, and Williams.

16) For enumeration of positroid cells of G+ and generating series interpolating between the Eulerians and Narayanaians, see A046802 and links therein.

17) Reflection group counting and q-counting, Reiner,

keywords: Catalan and Narayana numbers, parking functions, Weyl groups, q-extensions

Examining these brings to light another colorful thread in the tapestry related to moduli spaces, configuration spaces of particles, marked surfaces, and the polytopes associated to Lagrange inversion in different “coordinates”–o.g.f.s, e.g.f.s, etc. (The Lagrange inversion associated with the refined Eulerian partition polynomials seems to be “coordinate-free” in terms of the input in some sense.) Rather intriguing to me.

Refs added Nov. 2016:

18) Hedgehog Bases for A_n Cluster Polylogarithms  … , Parker, Scherlis, Spradlin, Volovich

19) Totally nonnegative Grassmannian and Grassmann polytopes, Lam

20) OEIS A248727: face-vectors of stellahedra / stellohedra, whose h-vectors enumerate positroid cells of the totally nonnegative Grassmannian. Related to the Eulerians.

21) http://4gravitons.wordpress.com/2015/07/31/amplitudes-megapost/

22) A131758 gives relations among the Eulerian numbers and polylogarithms of negative orders (see Wikipedia also).

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Compositional Inverse Operators and Sheffer Sequences

When considering operator inverses, one usually considers multiplicative inverses. As noted earlier in several entries, particularly, “Bernoulli and Blissard meet Stirling … ” (BBS), we see compositional inverse pairs of operators playing an important role in making associations among important integer arrays and combinatorics.

Using the arguments in BBS as a template, let

x = h(y)\;  and  y = h^{(-1)}(x), then, at any point (x,y) satisfying the inverse relations,

\frac{y}{x} = \frac{y}{h(y)} = \frac{h^{(-1)}(x)}{x}.

And, if these relations are satisfied about the origin, i.e.,

h(0) = h^{(-1)}(0)=0, and  h^{'}(0) =1, then these ratios serve as the e.g.f.s of the moments of the  Appell  sequences

\frac{t}{h(t)} \; e^{xt} = e^{tp.(x)}  and  \frac{h^{(-1)}(t)}{t} \; e^{xt} = e^{tq.(x)}.

Following the discussions in “Mathemagical Forests”, the e.g.f. of the binomial Sheffer sequence (u.(x))^n=u_n(x) associated to h(t), under these restrictions about the origin, is e^{u.(x)t} = e^{h(t)x}, and the lowering operator for the binomial sequence is L_u = h^{(-1)}(D_x) with D_x = d/dx.

Similarly, let e^{v.(x)t}=e^{h^{(-1)}(t)x}, and then L_v = h(D_x).

From the properties of such pairs of binomial Sheffer sequences, the umbral compositional inversion

u_n(v.(x))=x^n=v_n(u.(x)) also holds.

For any operator A, let

A^n = u_n(v.(A)) = u_n (B.)  with  (B.)^n=B_n= v_n (A).


D_A \; A^n = n \; A^{n-1} = n \; u_{n-1}(B.) = h^{(-1)}(D_{\omega}) \; u_n(\omega) \; |_{\omega =B.} \; =  h^{(-1)}(D_{B.}) \; u_n(B.) \; ,


\langle D_{B.} B.^n = n \; B.^{n-1}  \rangle \; = n \; B_{n-1} = n \; v_{n-1}(A) = h(D_A) v_n (A) \;

with \langle --- \rangle explicitly denoting the level at which the equivalent formal series  of reduced monomials of the umbral variable for the enclosed expression is to be umbrally evaluated.

In this sense, we obtain the pair of compositional inverse ops

D_A = h^{(-1)}(D_{B.})  and  D_{B.}= h(D_A)

and the relations

\frac{D_A}{D_{B.}} = \frac{D_A}{h(D_A)} = \frac{h^{(-1)}(D_B.)}{D_{B.}}.

We can relate this to matrix ops in the power basis  x^n through

D_x \; x^n =n \; x^{n-1}= D_x \; u_n(v.(x)) = \sum^n_{k=0} \; u_{n,k} \; \sum_{j=0}^k \; v_{k,j} \; j \; x^{j-1}, which implies that the lower triangular matrices of the coefficients of the two Sheffer sequences are a matrix inverse pair.

In addition, from the Appell formalism,

\frac{D_A}{h(D_A)} A^n = p_n (A) = \langle  \frac{h^{(-1)}(D_B.)}{D_{B.}} u_n (B.) \rangle = u_n(q.(B.))=u_n[q.(v.(A))]

and, conversely,

\langle  \frac{h^{(-1)}(D_B.)}{D_{B.}} B.^n \rangle = q_n(B.)= \frac{D_A}{h(D_A)} v_n(A) = v_n(p.(A))=v_n[p.(u.(B.))] \;,

giving conjugate relations among the two Appell sequences.

A particularly interesting example is when h(x) = e^x-1 with h^{(-1)}(x)=ln(1+x) , which is discussed in the earlier posts “Bernoulli, Blissard, and Lie meet Stirling and the simplices”  and “Goin’ with the flow,” related to the entry A238363.  Then the Bernoulli polynomials are given by

Ber_n (x) = \phi_n[q.((x).)]

where \phi_n (x) \; are the Bell / Touchard / exponential polynomials, or Stirling polynomials of the second kind; (x)_n \; , the falling factorial polynomials, or Stirling polynomials of the first kind; and the conjugated polynomials q_n(x) are presented in the OEIS entry.

This gives the formula for the Bernoulli numbers

Ber_n (0)= \sum_{k=0}^{n} \; (-1)^k \; St2_{n,k} \; \frac{k!}{k+1}  = \sum_{k=0}^{n} \; (-1)^k  \; \frac{Perm_{n,k}}{k+1} 

in terms of the Stirling numbers of the second kind or the coefficients of the face polynomials of the permutahedra / permutohedra (or dual polytopes, cf. A019538), e.g.,

\phi_4 (x)= \sum_{k=0}^4 \; St2_{4,k} \; x^k = x+7x^2+6x^3+x^4


Perm_4(x)= \sum_{k=0}^4 \; Perm_{4,k} \; x^k = x+14x^2+36x^3+24x^4.

Note the similarity of the expression for the Bernoulli numbers to that for the log of a determinant, or characteristic polynomial, in the MO-Q “Cycling in the zeta garden

\ln[det(I-uA_n)]=tr[\ln(I-uA_n)] = -\sum_{m\geq 0} \frac{tr(A_n^{m+1})u^{m+1}}{m+1} =-\sum_{m\geq 0} \frac{N_mu^{m+1}}{m+1} .

From discussions on the Pincherle derivative, if L is a lowering op for a sequence with the raising op R , then

D_L \; f (L)= [f (L),R].

Following the notes in BBS, Bernoulli Appells, and Goin’ with the Flow, all Appell sequences a_n (x) have the lowering op L=D=d/dx and a raising op of the segregated form R=x + H(D) and the logarithmic Appell sequence a_n (ln (x)), the lowering op L=xD and raising op R=ln(x)+H(xD) , so the commutator remains invariant to choice of the Appell sequence.

Another example of a (p,q) pair is provided by A132013 and A094587 with h(x)=x/(1+x) and h^{(-1)}(x)=x/(1-x), which is related to the Lah polynomials, or normalized Laguerrre polynomials of order -1.

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The Lagrange Reversion Theorem and the Lagrange Inversion Formula

From Wikipedia on the LRT, with

v(x,y) = x + y \; h(v(x,y)),

v(x,y) = x + \sum_{ n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x).

Letting x = 0 and w(y) = v (0,y),

w (y) = y \; h (w(y))  and h(y)=y/w^{(-1)}(y), giving

w(y) = \sum_{n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x) \; |_{x=0} \;,

the Lagrange inversion formula about the origin, whose expansion in the Taylor series coefficients of h(x) is discussed in OEIS A248927. See also A134685. For connections to free probability, free cumulants and moments, Appell sequences, noncrossing partitions, and other combinatorics, see A134264.

Let v^{(-1)}(x,y) be the inverse of v(x,y) w.r.t. to x . Then

x = v^{(-1)}(x,y) + y \; h(x), or

v^{(-1)}(x,y) = x - y \; h(x), and

v(x,y) = x + \sum_{ n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x) = x + \sum_{ n > 0} \frac {1}{n!} \; D_x^{n-1} [x-v^{(-1)}(x,y) ]^n.

The solution for the inverse of this last type is also presented in the post on the inviscid Burgers’ equation and the post  Generators, Inversion, and  Matrix, Binomial, and Integral Transforms.

The Laplace transform (LPT) argument in Appendix II of the Generators pdf can be briefly extended to derive the last form of the LRT above. (See below.)

Comparing v (x,t) with A(x,t) on page 2 of the Burgers’ equation pdf, we see that the the factor D^{n-1} \frac {h^n(x)}{n!} conjoined with y^n =t^n is equivalent to a summation over the t^n terms of all the partition polynomials (unsigned) in the expansion of A(x,t) on page 2 . Each partition polynomial is associated with a refined face polynomial for a Stasheff associahedron (cf.  MO-Q: … enumerative geometry and nonlinear waves?), or its dual, and the coefficient of the t^n term of the polynomial is a weighting of the (m-n+1)-dimensional face of the m-dimensional associahedron. For example, the t terms flag the full associahedron and the t^2 terms the facets, or the next lower dimensional faces, the (m-1)-dimensional faces of the m-dimensional associahedron. The summation for a given t^n then is a summation over the “column” space for the partition polynomials, representing a constant dimensional difference from the top-dimension of the polytopes.

The lead to the connection between the  OEIS entries and the LRT was Terry Tao’s post Another Problem about Power Series.

For comparison, in Tao’s notation the last equation here becomes

G(z)= v(z,y)-z = \sum_{ n > 0} \frac {y^n}{n!} \; D_z^{n-1} h^n(z) = \sum_{ n > 0} \frac {1}{n!} \; D_z^{n-1} [z-v^{(-1)}(z,y) ]^n
=\sum_{ n > 0} \frac {1}{n!} \; D_z^{n-1} F^n(z).

A formal derivation of the LRT:


f(x,t) = x -t \; F(x)= x - t \sum_{n>1} \; a_n \frac{x^n}{n!}


f^{-1}(x,t)= x + \sum_{n>1} \; b_n \frac{x^n}{n!}

be its compositional inverse in x. Then formally the Borel-Laplace transform gives, for a suitable class of functions,

1 + \sum_{n>1} \; b_n \; z^{n-1} = \int_0^\infty \frac{1}{z} \; \exp(-\frac {u}{z}) \; D_u(f^{-1}(u,t)) \; du =  \int_0^\infty \frac{1}{z} \; \exp(-\frac {f(u,t)}{z})  \; du 

=  \int_0^\infty \frac{1}{z} \; \exp(-\frac {u-tF (u)}{z}) \; du  = \sum_{n \geq 0} (\frac{t}{z})^n  \int_0^\infty \frac{1}{z} \; \exp(-\frac {u}{z}) \; \frac{F^n(u)}{n!} \; du.

Now inspecting the first expression in the chain of equalities shows that the terms b_n z^{n-1} need to be  multiplied by z/n! to obtain the terms of the formal Taylor series for f^{-1}(z,t), but this amounts to replacing z by 1/p, multiplying by 1/p^2, and taking the inverse Laplace transform, giving

f^{-1}(z,t) = z \; + \; LPT^{-1}_{p \to z} \left [\sum_{n > 0}  \frac{t^n}{n!} \; p^{n-1} LPT_{z \to p} \left [ F^n(z) \right] \right] = z \;+ \; \sum_{ n > 0} \frac {t^n}{n!} \; D_z^{n-1} F^n(z).

Do a sanity check with F (x)= x^m.

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Dirac-Appell Sequences

The Pincherle derivative [T^n(L,R),R] = \frac {d}{dL}T^n(L,R)= n \cdot T^{n-1}(L,R)  is implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, comprised of the Dirac delta function and its derivatives, formed by taking the inverse Laplace transform of the polynomials of an Appell polynomial sequence.

The fundamental D-A sequence can be defined as the sequence \delta^{(n)}(x) with L = -x and R = D = d/dx and e.g.f. e^{tD} \delta(x) = \delta(x+t). Another example is provided by OEIS A099174 with the D-A sequence H_n(x) = h_n(D) \delta(x) where h_n(x) are the modified Hermite polynomials listed in the Example section of the entry. The modified Hermite polynomials can be characterized several ways:

Continue reading

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The Pincherle Derivative and the Appell Raising Operator

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 ,


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 ,


[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 the dual relation

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

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

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).


Equation at top of page 5 should have 1/n rather than 1/n! in the series expanson.

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