,

with with , , and , construct the binomial Sheffer sequence with the exponential generating function

.

Then the associated formal group law (FGL) may be expressed as

The last operator factor may be expressed several ways:

where and .

The product of the Sheffer polynomials may be written, as discussed by J. Taylor in his thesis “Formal group laws and hypergraph colorings“, in terms of linear combinations of the Sheffer polynomials as

.

Then noting that is the lowering operator for the Sheffer polynomials, defined by , and that the raising operator, defined by , is (see Sec. 3: A Walk with Bruno and Blissard, pages 12 and 13, of my pdf “Mathemagical Forests” for derivations of these relations),

.

Evaluated at the origin,

where the umbral evaluation is carried out only after multiplying the polynomials together treating the umbral variable as a regular variable.

A good inverse pair for a sanity check is and for which the Sheffer polynomials are the falling factorials or Stirling polynomials of the first kind, . For example,

Note also that with and ,

so

and clearly if the coefficients of the formal Taylor series expansion of are non-negative integers, then so are the coefficients of . These coefficients are given in OEIS A145271.

My 2014 formulas in the OEIS entry for the Eulerian numbers A008292 give an FGL for . An extrapolated , which can be simply expressed as a polynomial, or infinite series, in the elementary symmetric polynomials/functions, gives the same in terms of the complete homogeneous symmetric polynomials/functions.

Explicitly, expressing in terms of the elementary symmetric polynomials and the complete homogeneous symmetric polynomials ,

Then

with

so

the Stirling partition polynomials of the first kind, a.k.a., the cycle index polynomials for the symmetric groups, described in A036039. See the link there to my pdf “Lagrange a la Lah” (particularly, pages 4 and 23 of Part I) for the first few polynomials and their characterization with as an Appell sequence in the indeterminate with the lowering and raising operators and

A related sequence of Appell polynomials in can be devised such that

where and

with for and . Giving for the first few,

Using the notation in Lagrange a la Lah,

Then

Note that

An umbral recursion relation follows from the last expression for the raising op:

The OEIS entry A263633 can be used to express the complete symmetric polynomials in terms of the elementary symmetric polynomials or vice versa. For easy reference for spot checks,

and

and the same holds for and interchanged.

Since , for ,

In addition,

Returning to the connection coefficients, we have

and

so

and

giving for the general FGL

This agrees with the particular

for the bivariate generating function of the Eulerian numbers A008292, for which for . (Here and are regular variables, not umbral variables. Note that a dot as a subscript is used to flag umbral variables in my notation.)

Spot checking with previous formulas, we have

and

since and

The coefficients of this last expression follow from A145271 with (cf. also A190015).

Note also, for , since

giving

For consistency between the operator formalism for umbral substitution and the direct replacement of an argument with the umbral variable, we must evaluate to . Then, since ,

The coefficients of the FGL remain invariant with a scaling of the indeterminates, i.e., with

Then and the associated are the refined Stirling polynomials of the first kind A036039, the refined Lah polynomials A130561, and the refined Bell polynomials A036040, respectively. The associated inversion partition polynomials for obtaining in terms of the same indeterminates are given by A133932, A133437, and A134685, respectively. Alternatively, A134264, A248120, or A248927 can be used for the inversion in terms of the coefficients of , for which calculation A133314 and A263633 are useful. Ultimately,

and except for .

Then can be used to give a recursion relation for each set of inversion polynomials in terms of the coefficients of the associated binomial Sheffer sequence and the lower order partition polynomials.

For example, consider A133437, the inversion formula for power series, or o.g.f.s, involving the refined face polynomials of the Stasheff associahedra, and A130561, the refined Lah polynomials, related to the elementary Schur polynomials. Given

the inversion polynomials of A133437 give

The associated binomial Sheffer polynomials as given by A130561 are the refined Lah polynomials

Then, e.g.,

or, equivalently,

And,

agreeing with the earlier result.

As another example, consider A134264, enumerating non-crossing partitions of polygons, giving the inversion partition polynomials in terms of the coefficients of the reciprocal

(The indeterminates here are not the complete homogeneous symmetric polynomials .)

Then

and, either from the e.g.f. or using the raising op

And consistently, gives

]]>

.

A generating function for this matrix is

with .

If , then also

,

where is a binomial Sheffer sequence of polynomials.

Then in this particular case,

and so is the product of an upper triangular Toeplitz matrix of divided-powers in , whose rows are the shifted summands of the Taylor series for , and the Sheffer polynomial summand matrix in . For example, these are the 4 by 4 submatrices:

.

By inspection,

where , the e.g.f. for the k-th column of the Sheffer matrix.

Revisiting the Lie infinigens of previous posts, we have, for and ,

,

and, consistently,

.

The trace for the general matrix,

with , appears in several guises (see the earlier post The Lagrange Reversion Theorem and the Lagrange Inversion Formula), changing colors but maintaining the same basic form, in related but distinct formulations for compositional inversion and, therefore, pops up in the analyses of formal group laws; antipodes for Hopf algebras; combinatorics of forests of tree graphs; convex polytopes; moduli spaces of marked discs and punctured Riemann spheres; Feynman graphs for quantum fields; solutions of nonlinear PDEs, such as the inviscid Burgers’ equation; Hirzebruch genera; and umbral, or finite operator, calculus.

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

,

where is to be regarded as a regular variable until the expression is reduced to monomials at which time it is to be evaluated as with , essentially the row polynomials of A094638 comprised of the Stirling numbers of the first kind. For example,

,

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 to be easily determined and factored out after the umbral evaluation.

Similarly, umbral reduction of transforms into the partitions of A248120.

“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.

,

so

,

and, with ,

,

with treated as independent of with respect to the derivations and, after being passed unscathed to the left of all derivations, is finally evaluated as .

Then, with ,

.

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

A generating function for the full set of JSPs is

.

This can be evaluated by noting, with

and ,

that

.

Then, for and , the operation reduces to

.

Then the generator gives, since ultimately ,

,

or

.

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

,

where acts as only on .

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

,

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

.

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 , 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.

Added 12/4/2016:

There are several other op reps for the JSPs.

,

so another rep is

,

,

where is the Kummer confluent hypergeometric function (see also earlier posts) and by definition . The reader can use the series expansion for the Kummer function in terms of rising factorials, the transformation , and the Chu-Vandermonde identity to confirm that

.

Added Dec 11, 2016:

Note that are the row polynomials of reversed, unsigned http://oeis.org/A049444 and reversed http://oeis.org/A143491.

]]>**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 ) : Totally non-negative Grassmannians ~ matroid polytopes ~ ~ degrees of toric varieties ~ number of solutions of polynomial equations related to scattering amplitudes in twistor string theory from volume/contour integrals over

II) : ~ 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 whose coefficient is the number of totally positive cells in that have dimension and goes on to show that for the binomial transform that , the Eulerians, and , 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 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**, (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**, (A033282, refined-A133437):

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

**Narayanaians**, (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**, (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.

**Questions**

**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?**

**References**

(More details for the interested.)

*Thread* :

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* ;

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).

]]>Using the arguments in BBS as a template, let

and , then, at any point satisfying the inverse relations,

.

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

, and , then these ratios serve as the e.g.f.s of the moments of the Appell sequences

and .

Following the discussions in “Mathemagical Forests”, the e.g.f. of the binomial Sheffer sequence associated to , under these restrictions about the origin, is , and the lowering operator for the binomial sequence is with .

Similarly, let , and then .

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

also holds.

For any operator , let

with .

Then

,

and

with 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

and

and the relations

.

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

, 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,

and, conversely,

,

giving conjugate relations among the two Appell sequences.

A particularly interesting example is when with , 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

where are the Bell / Touchard / exponential polynomials, or Stirling polynomials of the second kind; , the falling factorial polynomials, or Stirling polynomials of the first kind; and the conjugated polynomials are presented in the OEIS entry.

This gives the formula for the Bernoulli numbers

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.,

and

.

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”

.

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

.

Following the notes in BBS, Bernoulli Appells, and Goin’ with the Flow, all Appell sequences have the lowering op and a raising op of the segregated form and the logarithmic Appell sequence , the lowering op and raising op , so the commutator remains invariant to choice of the Appell sequence.

Another example of a pair is provided by A132013 and A094587 with and , which is related to the Lah polynomials, or normalized Laguerrre polynomials of order -1.

]]>,

.

Letting and ,

and , giving

,

the Lagrange inversion formula about the origin, whose expansion in the Taylor series coefficients of 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 be the inverse of w.r.t. to . Then

, or

, and

.

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 with on page 2 of the Burgers’ equation pdf, we see that the the factor conjoined with is equivalent to a summation over the terms of all the partition polynomials (unsigned) in the expansion of 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 term of the polynomial is a weighting of the (m-n+1)-dimensional face of the m-dimensional associahedron. For example, the terms flag the full associahedron and the terms the facets, or the next lower dimensional faces, the (m-1)-dimensional faces of the m-dimensional associahedron. The summation for a given 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

,

so the expansions in A248927 apply here as well with evaluation at rather than at the origin.

A formal derivation of the LRT:

Let

and

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

.

Now inspecting the first expression in the chain of equalities shows that the terms need to be multiplied by to obtain the terms of the formal Taylor series for , but this amounts to replacing by , multiplying by , and taking the inverse Laplace transform, giving

.

Do a sanity check with .

**Related stuff:**

1) “Formal groups, power systems, and Adams operators” by Buchstaber and Novikov

2) “The Hirzebruch criterion for the Todd class” earlier posting

3) “The methods of algebraic topology from the viewpont of cobordism theory” by Novikov

]]>The fundamental D-A sequence can be defined as the sequence with and and e.g.f. . Another example is provided by OEIS A099174 with the D-A sequence where are the modified Hermite polynomials listed in the Example section of the entry. The modified Hermite polynomials can be characterized several ways:

**A**)

**B**)

**C**)

**D**) .

The corresponding D-A sequence can be characterized similarly in various ways:

**A2**)

**B2**)

**C2**)

**D2**) .

See the MSE-Q “Extending an identity for the Dirac delta function” for a presentation of

,

which is useful for checking the identities.

The reps of the raising ops as conjugated versions of the iconic raising ops are easily understood in terms of umbral compositional inversion

.

As discussed in the Bernoulli Appells post, one generator for the Appell umbral compositonal inverse sequence is . For our Hermite polynomials, the umbral compositional inverse sequence of polynomials is given by (cf. A066325)

,

so

.

The basic Appell power series terms with raising op may be replaced by the basic D-A series terms with raising op and the same machinations hold.

Multiplying an Appell polynomial by another corresponds to convolving the corresponding D-A functions, which have representations as Cauchy complex-contour-integral operators and band-limited Fourier transforms, same as for the fractional calculus discussed in previous entries. For example, convolution of the basic D-A sequence gives

.

The relations in D2 can be expressed in other ways using basic properties of the Dirac delta function and identities in operator calculus:

where by definition and are the Bell / Touchard / exponential polynomials, discussed in several previous entries.

Inspecting the characterizations, we see that, given a basic sequence of functions with raising and lowering ops and , a generalized Appell sequence can be formed from a regular Appell sequence with and with

**A3**)

**B3**)

**C3**)

**D3**) .

So,

and

.

Be careful with the interpretation of the umbral operations here; in general,

and

unless ,

where is a non-umbral expresion independent of the value of , e.g., for the modified Hermite polynomials, but it could also be some operator that doesn’t act on .

An Appell sequence has the binomial shift property

,

but the associated generalized Appell sequence will not possess this property unless the underlying base sequence has it, i.e., unless . Only then will

.

In particular the Dirac-Appell sequence does not inherit this property since

.

Instead, the binomial convolution shift property for the basic Appell power basis translates into a multiplication by an exponential of the D-A sequence:

,

which can be confirmed by using the basic identity above for products of the power monomials with the derivatives of the delta function or through the inverse Laplace transform acting on with a simple translation of the complex integration variable .

Related Stuff:

“One-dimensional Quasi-exactly Solvable Schrodinger Equations” by Turbiner, p. 17

]]>

and

have the commutator relation

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

If for any particular natural number

,

then

,

implying

.

Since this holds for , the relation holds for all natural numbers, and formally for a function analytic about the origin (or a formal power series or exponential generating function)

.

The reader should be able to modify the argument to show the dual relation

.

This should be expected from representing the iconic operators in a dual Fourier space through which multiplication by becomes proportional to a derivation in the Fourier reciprocal space by, say , and derivation , to multiplication by .

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

The iconic ladder operators are and for the the powers , the prototypical Appell sequence of polynomials (see the post Bernoulli Appells for more on Appell sequences), so

,

and

.

If , then is the raising operator (see Bernoulli Appells) for an Appell sequence with moments given by the coefficients of the Taylor series for , i.e., ; lowering operator ; and e.g.f. .

Note that , so .

We’ve been using the power basis, but one should be able to construct a generalized Appell sequence from the raising and lowering ops of any sequence by letting and , where is a Taylor / power series about the origin or a formal exponential / ordinary generating function.

Added Oct. 2, 2016:

The Pincherle derivative is being 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 basic D-A sequence can be defined as the sequence with and and e.g.f. . Another example is provided by OEIS A099174 with the D-A sequence where are the modified Hermite polynomials listed in the Example section of the entry. The lowering and raising operators of this D-A sequence are and , and the e.g.f. is .

Multiplication of an Appell polynomial by another corresponds to convolution integrals of the corresponding D-A functions, which have representations as Cauchy complex-contour-integral operators and band-limited Fourier transforms.

**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/2015/11/21/the-creation-raising-operators-for-appell-sequences/

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

]]>

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).

**Errata**:

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

]]>**Additional notes**:

Using the inverse Mellin transform rep of the Dirac delta function given in an earlier entry leads to the integral kernel for on page .

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

,

which holds for any sequence of Appell polynomials and its umbral inverse Appell sequence . 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 :

,

so in this sense

.

**Errata:**

Pg. 2: should be .

Pg. 4: The lowering operator for the Bell partition and CIP polynomials gives , not the factor .

Pg. 9: should be .

**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.

]]>