Appell Matrices

Tom Copeland, Cheviot Hills, Los Angeles, Ca., Sept. 23, 2020

Each polynomial in an Appell sequence has the form

where I use the umbral notation and maneuver . I’ll refer to the as the base, or moment, sequence of the Appell sequence. The coefficients of the -th degree Appell polynomial are

The only restriction on the constants is that to ensure a canonical raising op for the sequence. The binomial coefficients in this identity are the elements of the Pascal matrix. (All matrices are lower triangular matrices in this presentation unless otherwise indicated.) They play a central role in the Appell formalism and in umbral calculus in general.

The associated exponential generating function for an Appell sequence is

with and the base/moment sequence

The coefficient matrix for the the Appell polynomials has the elements and, therefore, can be formed by multiplying the -th diagonal of the Pascal matrix by as in

Developing an infinitesimal generator (infinigen) for the Pascal matrix enables us to easily characterize the inversion and product of the coefficient matrices of Appell sequences and to generalize the formalism to factorial matrices other than the Pascal. The infinigen for the Pascal matrix is, of course, an infinite matrix, but the truncation of the infinigen to rank (the n-jet of the infinigen, so to speak) is nilpotent of order , allowing us to deal only with finite matrices, or, equivalently, a graded algebra for the first polynomials of an Appell sequence.

We could take the log of the Pascal matrix to obtain its infinigen, but to elucidate the connections to a broader calculus (or indeed easily justify taking the log), let's approach the infinigen via differential operators. The action of the derivative on a polynomial lowers the degree of the polynomial by one. The rank four matrix, , representing the operator for action in the power basis on the polynomial is, with zeros above the diagonal suppressed,

Since repeating the differentiation again and again eventually annihilates the polynomial, the truncated infinigen rank must be nilpotent of order , i.e., in this instance, . Squaring the matrix gives the operator for twice differentiating the polynomial (or a 3-jet)

and cubing (or thrice differentiating),

We obtain the rank 4 submatrix of the Pascal matrix with the exponential

The truncated Pascal matrix at any rank is generated in this manner by the infinigen with the only nonvanishing diagonal the first subdiagonal . The same is true for any factorial matrix, i.e., one with an infinigen with the only nonvanishing diagonal the first subdiagonal , so I want to present the results for the Pascal matrix in more general notation. The nonvanishing -th subdiagonal of is

up to the last element terminating with the factor , a sliding product of equal number of factors for a given subdiagonal--one factor for the first subdiagonal; two, for the second; three, for the third, etc. just as we obtain for successive differentiation of .

For the Pascal matrix, the exponential of this infinigen then has elements that can be represented as the binomial coefficients with the convention so that the binomial coefficient vanishes for . For generalized factorial matrices, an umbralized binomial coefficient can be used to represent each element concisely as

vanishing for . (For an even more generalized binomial coefficient, see the Narayana numbers A001263.) ​ ​

Inspecting the binomial convolution formula in the introductory sentence, we see that the coefficient matrix for the first polynomials of an Appell sequence can be expressed as

Therefore, multiplication of the coefficient matrices of two Appell sequences of rank is commutative and given by

This consistently translates into the umbral composition of the Appell polynomials

where the resulting Appell sequence has the base/moment sequence

i.e., the binomial convolution of the base/moment sequences of the two Appells, which can also be obtained by multiplication of the e.g.f.s of the moment sequences

Then the e.g.f. for the product of the coefficient matrices of two Appell sequences, or, the umbral composition of the two sequences is given by

Now introduce a dual Appell sequence , the umbral compositional inverse (UCI), defined by the property

implying for ,

This gives an intertwined recursion relation for the two sequences.

The UCI relations also imply the e.g.f. identity

which is the e.g.f. of the row polynomials of the identity matrix , the coefficient matrix for the prototypical self-umbrally-inverse Appell sequence of simple monomial powers for which . Consequently, the moment e.g.f.s of an Appell sequence and its unique Appell UCI are a pair of multiplicative inverses; i.e.,

and, equivalently (when considering only Appell coefficient matrices), the Appell coefficient matrices of the duo are a matrix inverse pair, i.e.

Reprising, the group of Appell sequences is closed and Abelian under umbral composition, and the umbral composition of two Appells has the three equivalent reps

1) polynomials: ​ ​

2) e.g.f.s: ​ ​

3) coefficient matrices: ​ ​

with comprised of the binomial convolutions .

The UCI relationship has the three manifestations

1) polynomials are umbral compositional inverses: ​ ​

2) moment e.g.f.s are multiplicative inverses: ​ ​ ​ ​

2) coefficient matrices are multiplicative inverses: ​

with the column vector with components resulting from the binomial convolutions .

There are more relations presented in OEIS A133314, including connections to permutahedra ​ ​ and surjective mappings, and in previous posts on the Appell and other Sheffer sequences. One that I have alluded to in several OEIS entries, but not elsewhere, is that between production ​ matrices for Riordan arrays and matrix reps of the raising operator for an Appell sequence ​ incorporating and its powers. The raising op is defined by

A derivation of diff op reps of follows from the interplay of the Appell sequence and its umbral inverse. With ,

so

implying

The last two equalities can be derived from the Pincherle derivative as in my posts "The ​ Pincherle Derivative and the Appell Raising Operator" and "Bernoulli Appells" and “The ​ ​ ​ ​ Creation / Raising Operators for Appell Sequences”, but here's yet another proof for the reduced ​ conjugation:

Since the lowering op for Appells is the derivative op, i.e., , the action of the commutator on the power polynomials is

so indeed

More than one way to skin a cat though. Here’s an earlier, simpler derivation..

therefore,

so for an analytic function

Finally, the production matrix mentioned above is the transpose of the matrix rep for the raising operator. The more general Graves-Pincherle derivative for any two associated lowering and ​ ​ raising operators and for a polynomial/function sequence is

Other factorial matrices can be found in OEIS (search for infingen in the OEIS). For example, the Lah matrix of OEIS A105278, a generalized factorial matrix whose infinigen is A132710, can ​ ​ ​ ​ be used to generate the generalized Appell polynomial sequence with the coefficient matrix

Inverting and extracting diagonal constants gives

This is a graded algebra, and contains the first four coefficients of the formal Taylor series of . (I say formal since as a graded algebra, it works for the formal e.g.f. of divergent series equally well.) The partition polynomials are the general Euler characteristic classes for the permutahedra (see A133314 for more on this). ​ ​

Since and are an inverse pair, the pair of polynomial sequences represented by their coefficients are an UCI pair. These are not Sheffer polynomial sequences but can be related to the Lah binomial Sheffer sequences.

For an infingen that does not generate a factorial matrix, see A238385. For infingens, for the ​ ​ associated Laguerre polynomials, examples of general Sheffer sequences emerging from a binomial Sheffer sequence, see A132681. Several of my other posts that have Infinigen, Appell, ​ ​ or raising op in the titles contain related material.

The representation of a matrix as where is a nilpotent infinigen with only the second subdiagonal nonvanishing allows for the definition of generalized Appell sequences in terms of a base/moment sequence with e.g.f. and the coefficient matrix in the power basis

These polynomials share some of the properties of the canonical Appell sequences; for example, given the infinigen and the -th degree polynomial, all the lower degree polynomials are easily determined through the diagonal multiplication illustrated above that the coefficient matrix manifests. In addition, the umbral composition properties are isomorphic since

If we change the basis set from. say, the powers to the divided power , we have a different set of infinigens with coefficient matrices formed by

and convolutions of moment vectors defined by ordinary generating functions rather than e.g.f.s;

where and the convolution of the two vectors is defined by multiplication of o.g.f.s, i.e.,

For example, the infingin associated with differentiation of divided powers in the divided powers basis has the nonvanishing subdiagonal of all ones reflecting . For more on this, see “Infinigens, the Pascal Triangle, and the Witt and Virasoro Algebras.” ​ ​

The recent MathOverflow question and answer “Are any interesting classes of polynomial ​ sequences Sheffer sequences groups under umbral composition?” presents some discussion ​ ​ of umbral composition along with a link to some generalizations in “Generalized Riordan groups ​ and operators on polynomials” by Zemei. ​

(By the way, umbral as coined by Sylvester doesn't mean shady as in the pop math meme spin on this term but rather a shadow of binomial convolution in its several avatars, a shadow of simpler algebraic maneuvers.)

Last but not least, let’s consider umbral composition in general and its relationship to matrix algebra regardless of whether there is an underlying fundamental matrix generated by an infinigen. Consider two sequences of polynomials as above whose coefficient matrices are lower triangular matrices. The umbral composition is equivalent to matrix multiplication of the two coefficient matrices

and

can be identified as elements of the coefficient matrix

This is in general not a commutative product and so neither is the umbral composition an Abelian operation, in contrast to the group of Appell polynomials, unless the two matrices are an inverse pair, i.e., the two sequences are an umbral inverse pair.

We can conclude, for umbral inverse pairs of polynomial sequences characterized by lower triangular coefficient matrices, regardless of their origin, that

Is equivalent to

and that, for any two polynomial sequences characterized by lower triangular coefficient matrices, umbral composition

is equivalent to the ordered product of the coefficient matrices