Reciprocity and Umbral Witchcraft: An Eve withStirling, Bernoulli, Archimedes, Euler, Laguerre, and Worpitzky

Tom Copeland, Cheviot Hills, Los Angeles, Ca., Aug. 10-21, 2020

Motivated by the appearance of the Eulerian polynomials in algebraic geometry, geometric combinatorics, and in some derivations of the Baker-Campbell-Hausdorff-Dynkin (BCHD) expansion, identities are generated using umbral Sheffer calculus couplings of the iconic

inverse pair and , naturally relating the Stirling polynomials of the first and second kinds; the Eulerian, the Bernoulli, and the generalized Laguerre polynomial sequences; the canonical and umbrally morphed Worpitzky identities (WI); and the combinatorics of the classic simple convex polytopes the permutahedra/permutohedra. Relatively new polynomial sequences are procreated as well and a not oft-noted genealogy drawn encompassing the Kummer confluent hypergeometric polynomials, the Swiss-knife polynomials, the Worpitzky triangles, and the f- and h-polynomials of the stellahedra (whose h-vectors enumerate the positroids, or non-negative Grassmannians). These polynomials, their e.g.f.s (or related o.g.f.s), their associated generalizations and differential ops and identities, or similar, are often found in presentations on symmetric functions, quantum field theory, combinatorial Hopf algebras, L-function theory, characteristic classes, volumes of polytopes, and the Magnus expansion and other methods of Lie/geometric numerical integration.

These notes first give birth to a sextuplet of Sheffer polynomials from a natural coupling of two

parent functions and . This sextuplet is an extension of the quadruplet of an earlier post "Compositional Inverse Operators and Sheffer Sequences" in my WordPress math blog Shadows of SImplicity. The quadruplet plays the major role in these current notes. Any pair of compositional inverses (or 'reciprocal') functions for which may serve as ‘parents’ procreating a sextuplet of the general kind produced here. Of course, the genetics may be more interesting or less than that of this particular family.

From the embraces in a dance among the sextuplet issues two basic dual polynomial identities one for the Bernoulli polynomial and the other for its umbral inverse sibling the Reciprocal polynomials. From the basic Bernoulli polynomial identity further identities are derived--first for the Bernoulli numbers which then serve as guides to further general Bernoulli polynomial identities. Umbrally morphed Worpitzky identities embracing the Eulerians also organically spring forth, identities that can be found in earlier posts and in my even older OEIS contributions.

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‘Witchcraft’, ‘child’s play’, wonder at simple, elegant, powerful patterns at play--natural magic.

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The umbral embrace of a bewitched sextuplet "As a matter of fact, the feeling of witchcraft that has hovered over umbral calculus is probably what kept it from dying altogether. ... At long last, it was realized that umbral calculus could be made entirely rigorous by using the language of Hopf algebras, and this was done in a lengthy treatment. However, although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of 'classical' umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside." (Gian-Carlo Rota, 1994).

Let’s conjure up Blissard and some conjugal umbral magic with a sextuplet of Sheffer sequences--two binomial sequences, and , and four Appell, and and

and --via two binomial e.g.f.s

and four Appell

The binomial Sheffer sequences are an umbral compositional inverse pair (see Appendix III) and so are two pairs of the Appells and (see Appendix I); that is,

Focusing on the binomial and the first pair of Appells , we have

and the general Sheffer sequence a semiproduct of an Appell and a binomial sequence)

so

Repeating these umbral maneuvers with the binomial and the second pair of Appells , we arrive at the dual pair of umbrally conjugated polynomial identities (UCPIs)

A quick check of the consistency of the first three unitary and the UCPIs is embraced in

Written, for the uninitiated and those who abhor witchcraft and for quick numerical checks, in terms of the identity matrix and lower triangular coefficient matrices for the sequences of polynomials, the equations translate into the unitary

the conjugations

and the consistency check

Now specialize for some Bernoulli UCPIs (BUCPIs). Choosing the parent , we have the two Sheffer binomial sequences, the Stirling polynomials of the second and first kinds

and four Appell, including the Bernoulli polynomials and its umbral compositional inverse sequence the Reciprocal polynomials ,

and the BUCPI and the dual Reciprocal UCPI

(Spot check these with the low rank polynomials in the Appendices--most easily done with their coefficient matrices, but do a couple using umbral composition to develop some facility with umbral maneuvers.)

These equations relate the Bernoulli polynomials to the Appell polynomials (see Appendices I & IV) through a similarity transform/conjugation by the Stirling polynomials of the first and second kinds , a.k.a. the Bell polynomials , (see Appendix III), and, since

the BUCPI reduces for to a Bernoulli number identity (BNI)

where are the canonical Bernoulli numbers. We have just conjured up a manifestation of the Archimedean solid the truncated octahedron and its hyper-dimensional polytope analogs the simple convex permutahedra, for their augmented (face) f-polynomials , their shifted reverse f-polynomials , and their augmented h-polynomials, the Eulerian polynomials

(see Appendix II) are related as

and thence to the Stirling numbers of the second kind via

Hence,

Consequently, with

,

we have

and, for , the permutahedra connection reduces to the BNIs

and, using

we have

implying, for ,

Reprising, we have the BNIs,

Since the Bernoulli numbers vanish for all odd indices greater than 1, for which , and vanishes for , this is in agreement for with the Wikipedia under Eulerian

numbers, with ,

Because of the similarity in the e.g.f.s of the various number arrays, a derivation, but not geometrically motivated, can also be obtained by massaging the e.g.f.s.

Let’s look at another derivation using similar steps.

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A variation of the embrace

Return to the Bernoullis,

Transform this with the inverse to get

or

where

Then since the Bernoulli polynomials, as do all Appells, have the shift property, the binomial expansion gives

or, equivalently,

With umbral composition, this becomes

and the less elegant, trickier relation

Conjure up once more the Archimedean permutahedra, and we have again

or, changing signs,

Integrating from to ,

Equating the two identities for gives our identity in the previous section

Next, we look at a clan of Laguerre polynomials, embrace the canonical and umbrally morphed Worpitzky identities, and extend the Bernoulli polynomial identities.

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A broader embrace (manifesting Worpitzky identities)

Conjuring up the permutahedra connection again,

and, consequently,

or

Then

introducing the associated Laguerre polynomials

(Befitting the entire reciprocal Euler gamma function, I use the convention for when necessary.)

Invoking the Chu-Vandermonde identities (alternatively see the MO-Q Pocchammer symbol of a differential and the hypergeometric polynomials),

implying after composing with that

giving the Bell morphed Worpitzky identity,

By umbrally composing this with , we morph back to the canonical WI,

and, umbrally morphing again,

Umbrally composing the Bell-Laguerre-Worpitzky identity above gives

and composing once more with , we now have a consistent extension of the basic BUCPI to include the Eulerians and Laguerre polynomials

Note that umbrally substituting the Reciprocal polynomials , the UCI of the Bernoulli polynomials, into this last set of equations generates the Worpitzky identities again, or, equivalently, acting on it with the op

consistent with our sextuplet interplay

and

Evaluating the BUCPIs at and comparing with the previous BNIs,

(The generalized Chu-Vandermonde identities really lie at the heart of the relations among the Kummer confluent hypergeometric/generalized Laguerre functions and their inverse Mellin-Barnes contour integral reps.)

These are a few steps in a larger dance.

Reprising the Bernoulli number identities (BNI) from above,

and

Then suddenly a new manifestation, a familiar hidden in the shadows by the glade all along, of one of our shy Appells, kin to the harmonic numbers and digamma ,

and an invitation to variations of the dance once more.

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The power of suggestion

Umbral witchcraft wonderfully conjures the more general from the simpler.

Let’s dance with the familiar above

The diff op (as Jeans would say) is hungry to feed on something. Let it act on with first. Then

We could umbrally compose this with , or equivalently, let the diff op act on

but with this time. Then

Again we have a cameo by Chu and Vandermond with

We know by the BNIs above that for a natural number these expressions reduce to , and our familiar is kin to the finite difference operator, so recast the identities using

the shorthand

Now

for , where are the generalized harmonic numbers (e.g., see this MO-Q & A “Lie group heuristics for a raising operator” and my posts on the digamma function). Then we have our general identities

We could dance with this clan all night, but let’s change partners, bowing out to Riemann with a final few moves: for for some natural number ,

and composing with and recalling its derivative property, we have

which we shall see is consistent with some more general derivations.

Another illustration of the power of suggestivity of umbral witchcraft: The initial BUCPI involves the Appell polynomials , which may be obtained (see Appendix I) using a diff op formed

from their e.g.f. as

where

Then acting on the Bell polynomials with this op--this is a demonstration and justification for umbral substitution/composition--gives

Evaluating at and using the BNIs above, gives

leading us to surmise the more general

and, by umbral substitution, the extension of the BUCPI

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A powerful charm

Double, double toil and trouble;Fire burn and caldron bubble.

Cool it with a baboon's blood,Then the charm is firm and good. -- Shakespeare

So, let’s stir our vat of Appells and add some spices to conjure up some more manifestations of the Bernoullis. Let’s umbrally morph our BUCPI

into

and normalize the dual

into

Using the shift property of the Appells, it is also true that

which umbrally morphs into

Now going general again and repeating our earlier relation

but also

so

and

giving

Equivalently,

and

This reverberates back to the Bernoulli sextuplet, manifesting as

or

or

where

Let’s corroborate this with another derivation while nurturing some more insight into related diff

ops. Noting that since the Bernoulli polynomials are an Appell sequence with e.g.f. ,

(confirming from Appendix III that is the lowering op for )

so

or, equivalently,

where are the signed row polynomials of A028421, since the e.g.f. for is given by

Note again that the form of the e.g.f. is that of a general Sheffer sequence, a semidirect product of an Appell and binomial Sheffer sequence, or, more precisely, the lower triangular coefficient matrix for is given by , i.e., umbrally, in

agreement with A238363 (see Appendix IV). Consequently, is the lowering op for the sequence as well as for the ; i.e.,

or

By the properties of Appell polynomials and derivatives,

Also, using Euler's reflection formula for the gamma function,

which evaluated at , gives

so we have again, for ,

consistent with the BNIs above.

Returning to the more general relation and evaluating umbrally, we get

from the earlier BUCPI. This agrees with the definition of based on operator equations in A238363 (see Appendix IV). And umbrally substituting in , we have

specializing for any integer to

There are operational and combinatorial derivations of most if not all of the relations above perhaps worth looking at in more detail (or reviewing and collating), and certainly there are other interesting sextuplets to dance with, but that’s for another eve in the forest.

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Appendix I: The Bernoulli numbers, polynomials, and (Todd) diff op

(This serves as a review of some properties of Sheffer Appell sequences of which the Bernoulli polynomials are historically the initial prototype after the basic power monomial sequence .)

The Bernoulli numbers begin as

Because of some gratuitous arbitrariness among some authors in defining the Bernoulli numbers, I'll define them in the way consistent with naturally occurring operational identities and diff ops as in my last post "Differintegral Ops and the Bernoulli and Reciprocal Polynomials" as, with ,

Then with , we find the e.g.f. of the Bernoulli polynomials is

giving . (Some choose , which corresponds in terms of the Bernoulli polynomials to rather than . The diff ops

are sometimes called the finite difference operator and the Todd operator, respectively, and naturally occur in derivations of the Euler-Maclaurin, BCHD, and Magnus expansions and other Lie theoretic analyses. They are central to the development of the finite difference calculus and play significant roles in algebraic-geometric combinatorics.

The first few Bernoulli polynomials

are

(Always good to check the polynomials with the lowering op . )

The Bernoulli polynomials are an iconic Appell Sheffer sequence of polynomials. In showing and deriving some properties of Appells and associated diff ops, I’ll use the convenient and suggestive elegance of umbral notation and designate umbral characters with a dot as a subscript, .e.g., . It is to be understood that the lowering of the exponent to the index occurs only after a side of an equation has been reduced to a unique power series in the umbral quantity, i.e., to a sum of the monomials . Where umbral evaluation occurs at a separate level (not needed here), I would use an expression like

which is generally not equal to

Any sequence of numbers beginning with 1 can be used to define an Appell polynomial

sequence. Call , with the e.g.f. , the moments of the Appell polynomial sequence. Then the Appell sequence can be defined as

so

and

A dual Appell sequence defined by the reciprocal of the e.g.f. of the moments is the umbral compositional inverse of the sequence . Unpacking this assertion, let

Then

and

The central operation is commutative, so we conclude the pair of Appells are an umbral compositional inverse pair; that is,

(How abou’ dem Appells?!) Well, there is more to the story than this (see my other posts on the Bernoullis), but we have sufficient weaponry to attack our problems now.

The Bernoulli Appell polynomials are defined by their moments, the Bernoulli numbers with

e.g.f. ; and their dual, the Reciprocal polynomials , by their moments, the

reciprocal natural numbers with e.g.f. .

As mentioned earlier, the Bernoulli numbers and their elegant consorts, the reciprocal natural numbers appear in diverse mathematical landscapes from algebraic and geometric combinatorics, algebraic geometry, characteristic classes and differential geometry, the differential/integral and finite calculi, quantum field theory, to number theory, in particular, the Riemann zeta function (see. e.g., “Topological Methods in Algebraic Geometry” by Hirzebruch and my posts on my blog.)

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Appendix II: The Eulerians, f- and h-polynomials of the permutahedra, and associated polynomial and number sequences

There are three common versions of the Eulerian polynomials that will be useful to our purposes:

I) A123125

with e.g.f.

which begin as

II) A173018 , the reverse of the Eulerian polynomials above, i.e.,

with e.g.f.

which begin as

III) A008292, the h-polynomials for the permutahedra with subscript matching the dimension of the polytope

which up through the third dimension are

Note the palindromic, or reflection symmetry, typical of the h-vectors of simple convex polytopes.

They are related to several other important number sequences. Construct an Appell sequence

in the variable with as the moment sequence and

with e.g.f.

Then with ,

1) are the Eulerian polynomials A123125, the shifted h-polynomials of the permutahedra / permutohedra

2) are the h-polynomials of the stellahedra A046802, which also enumerate the positroid cells of the totally non-negative Grassmannians,

3) are the (face) f-polynomials of the stellahedra A248727,

4) are the Swiss-knife polynomials of A119879,

5) are the row polynomials of the Worpitzky triangle A130850,

6) are the row polynomials for the reverse Worpitzky triangle A028246, a specialization of the logarithmic partition polynomials A263634,

7) are the h-polynomials of the permutahedra A008292,

8) are the face or f-polynomials of the permutahedra A090582,

augmented with an initial 1 and with subscript one higher than the dimension, with e.g.f.

and the first few of are

9) are the reverse f-polynomials of the permutahedra A019538 with umbral refinement https://oeis.org/A133314, associated to multiplicative inversion of e.g.f.s. Their e.g.f. is

and the first few of are

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Appendix III: The Stirling polynomials of the first and second kinds

Both sequences lie at the crossroads of important combinatorial, algebraic, operational, analytic, and geometric paths-- even more so the generalized versions, the Stirling partition polynomials of the first (A036039) and second (A036040) kinds.

The Stirling polynomials of the first kind are the lowering factorial polynomials defined by

or operationally by

where, for convenience I define These are closely related to the Pochhammer symbol, a.k.a. the rising factorials.

The Stirling polynomials of the second kind can be defined operationally as

and, therefore,

and, conversely,

so

i.e., the pair are an umbral compositional inverse pair.

The Stirling polynomials of the first kind are also known as the Bell / Touchard / Exponential polynomials, and I use the alternate notation for them here as well.

The form of the e.g.f.s of this pair of polynomial sequences imply they are binomial Sheffer sequences. The e.g.f.s of the Stirling polynomials of the first kind (here we use A048994, other versions are A008275 and A094638) and second kind (here we use A048993, another version is A008277) are

They begin with

and

We can identify these two binomial Sheffer sequences as an umbral compositional inverse pair

from their fundamentally defining functions and being a functional compositional inverse pair. This is most easily seen from the umbral equation

which, naturally, also holds with the polynomials switched. This also implies the lower triangular coefficient matrices of the pair are a multiplicative inverse pair.

There are also the associated pair of lowering operators

and

These follow from the relations for general binomial Sheffer sequences

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Appendix IV: The Appells q_n(s) and c_n(s)

The Appell polynomials , operationally defined in A238363 (signed reverse of A111492) , have the the e.g.f. and moments

The sequence

begins with

where the numerators are the polynomials of A238363, whose e.g.f. is but the shifted polynomials of A238363 are not an Appell sequence. (Spot check with

). The can be defined operationally in a few ways; for example, it’s an Appell sequence, so as discussed in Appendix I,

I stumbled onto this sequence by looking at a Pincherle-Lie bracket giving

or

equivalent to

where are the signed row polynomials of A028421, the derivatives of the unsigned Stirling polynomials of the first kind.

Then the e.g.f. for is

(This is a brief repetition of a derivation of the e.g.f. In the main text, and since this is a derivative, we must have, by the operational definition of the Bernoulli polynomials, a relation as in the main text to the Bernoulli polynomials.)

The row polynomials of signed A028421 begin with

To illustrate the calculations (and as a spot check):

so

confirming that

For the corresponding operator calculations, we need to unpack

From Appendix III,

so

which we have calculated explicitly above with and replaced by in the first and final expressions.

Note that the coefficient matrix of A238363 is a conjugation of the infinitesimal generator (cf. A132440 ) of the lower triangular Pascal matrix by the Stirling matrices

so its row polynomials are given by

where, for ,

In other words, the elements of are zero except along the first subdiagonal, which begins as .

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Appendix V: The row polynomials

I hadn’t really come across these row polynomials operationally or otherwise before writing up these notes and don’t see them in the OEIS, but I wouldn’t be surprised if they are somewhere in Norlund’s work. I did find some aspects of them.

The polynomials have the e.g.f.

and begin with

The denominators up to order 8 coincide with A091137 and are

They can be found in “Intégration numérique des systèmes différentiels à conditions initiales” (Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969) by Curtz and in the Todd polynomials on page 14 (and, according to Sloan, is calculated by Lemme 1.7.3 ) of “Topological Methods in Algebraic Geometry” by Hirzebruch. Hirzebruch mentions that a simple transformation relates the Todd to the generalized Bernoulli polynomials of Norlund, formed by binomially convolving the Bernoulli numbers, or, equivalently, umbrally composing the Bernoulli polynomials with themselves or repeatedly multiplying the moment e.g.f. by itself (see my post “The Hirzebruch Criterion and the Todd Class”). Also see A091137 for a conjecture that the sequence is related to reversion of the dilogarithm, or , the polylogarithm of order two. This is not so surprising given connections drawn in my entry A131758 of polylogarithms to the Eulerian numbers and pretty much every other sequence in this paper. A partial sequence also

appears in https://oeis.org/A165313.

The constant terms of the numerator polynomials up to order 9 coincide (mod signs) with https://oeis.org/A141417 (with Curtz again as a reference) and are

.

Both the numerator and denominator sequences, as well as the other Appell polynomials in the sextuplet, are related to numerical/Lie/geometric integration

The ratios formed from the numerator and denominator sequences are, of course, the moments (a.k.a. the derivatives at the origin, or Taylor series coefficients) of .

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Appendix VII: Miscellaneous comments on the sextuplet

Repeated convolutions of each of the Appell polynomials of the sextuplet can be achieved by

repeatedly operating on its e.g.f. with forming in some sense a ladder of “tangent” spaces. Norlund relates these generalized polynomials for the Bernoulli numbers to expressions for higher order derivatives, and Hirzebruch, to his Todd classes. We see how a conjugation by the Stirling matrices relates the Bernoulli polynomials to the q-polynomials, so their umbral self-convolutions must also be related to higher order differentiation. The dual umbral conjugated polynomial identity (RUCPI) imply the same is true of the other pair of Appells in the sextuplet, which are the respective reciprocals, or umbral compositional inverses, of the pair.