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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