**UMBRAL NOTATION** (courtesy of Blissard)

$$(q.)^n = q_n$$

Concise, elegant, and suggestive, it often allows for both brevity and comprehensibility of presentation and short derivations of operational results.

**Examples of umbral variables**:

$q.^n = q_n = d$, a constant quantity, maybe a single variable or polynomial,

$q.^n = q_n = d^n $, a constant quantity raised to powers,

$q.^n = q_n = x^n$, a variable raised to a power,

$q.^n = b.^n = b_n$; an element of a sequence of numbers, such as the Bernoulli,

$q.^n = B.(x)^n = B_n(x)$, an element of a sequence of polynomials, .e.g., Bernoulli,

$q.^n = (.)!^n = (n)!$, an expression containing an integer parameter,

$q.^n = \binom{x}{.}^n = \binom{x}{n} $, ditto,

$q.^n = M.^n = M_n$, a sequence of matrices, differential ops, ... whatever.

**The umbral maneuver in more complex expressions**:

reduce an expression that contains the umbral entity to an analytic series in monomial powers of $q.$ when possible

then lower the positive integer exponent to a subscript

when in doubt apply

$$e^{q.xD_y} \; f(y) \; |_{y=0} = f(q.x +y) \;|_{y=0} = f(q.x) \;$$

- sensible results can often be obtained for functions not analytic at the origin from

$$f(q.x) := e^{-(1-q.)x D_y} \; f(y) \; |_{y=x} = f(x-(1-q.)x) = f((1-(1-q.))x) $$

(See below on Newton interpolation. E.g., with $f(x)=x^s$, then $f(q.x) = (q.x)^s = q_s\; x^s \; := (1-(1-q.))^sx^s \; $ when convergent.)

**Basic umbral operations**:

- binomial convolution for commuting quantities:

$$c_n = c.^n = (a. + b.)^n = \sum_{k=0}^n \binom{n}{k} \; a.^k \; b.^{n-k} = \sum_{k=0}^n \binom{n}{k} \; a_k\; b_{n-k},$$

- formation of e.g.f.s and o.g.f.s:

$$E(t) = e^{q.t} = \sum_{n \geq 0} \; q.^n \; \frac{t^n}{n!} = \sum_{n \geq 0} \; q_n \; \frac{t^n}{n!}$$

and, with $q_0 =1$,

$$O(t) = \frac{1}{1-q.t} = \sum_{n \geq 0} (q.t)^n = \sum_{n \geq 0} q_n \; t^n = e^{\bar{q}.t}= \bar{O}(t)$$

with $\bar{q}_n = n! \; q_n$.

Then multiplication of formal series is

$$EA(t) \; EB(t) = e^{a.t} \; e^{b.t} = e^{(a.+b.)t} = e^{c.t} = EC(t) \; .$$

We can even disregard convergence of both series if we are interested only in formal compositional or multiplicative inverses or the formal Laplace transform of an e.g.f. to an o.g.f. since the production of the $n$-th coefficient of the target series depends only on the coefficients of order $n$ or less of the source series for these transformations.

**Brevity and intelligibility in umbral versus conventional presentations**:

The Euler transformation for e.g.f.s in umbral notation

- $$ e^{a.x} = e^x \; e^{-(1-a. )x}$$

versus conventional notation

- $$ \sum_{n \geq 0} a_n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} (-1)^n \; [\; \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} \; a_k \; ] \; \frac{x^n}{n!} \; ,$$

and derivation of the conventional formula via the umbral,

- $$ e^{a.x} = e^x \; e^{-x} e^{a.x} = e^x \; e^{-(1-a. )x}$$

$$ = e^x \; \sum_{n \geq 0} (-1)^n \; (1-a.)^n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} \; (-1)^n \; [\; \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} a_k \; ] \; \frac{x^n}{n!} \;.$$

(23 keys tapped for eqn. 1 versus 102 keys for 2, not counting the appreciable difference in taps for blank spacing and latex spacing for readability. I could have saved a good fraction of a lifetime of math work by being able to use umbral notation in symbolic apps, study, and publications rather than sigma gymnastics.)

**Natural generalizations via umbralization--monic to partition polynomials:**

Stirling polynomials of the first kind, related to cyclic permutations, have the e.g.f.

$$e^{St1.(x)t} = e^{x \ln(1+t)} = (1+t)^x\; ,$$

and Stirling polynomials of the second kind, related to partitions of sets,

$$e^{St2.(x)t} = e^{x (e^t-1)} \, .$$

Flexibility and precision is afforded if some notation is used to indicate at what level the umbral exponent lowering is to occur. For example, with $\langle \cdots \rangle$ denoting evaluation of the expression between the angular brackets after reduction to a series in the umbral quantity and with $q_0=1$,

$$\langle \; e^{-\ln(1-q.t)} \;\rangle = \langle \frac{1}{1-q.t}\rangle = \langle \sum_{n \geq 0} q.^n \; t^n \; \rangle = \sum_{n\geq 0} \langle q.^n \rangle \; t^n =\sum_{n \geq 0} q_n \; t^n $$

and

$$e^{-\langle \ln(1-q.t)\rangle} = \exp[\langle \sum_{n \geq 1} q.^n \frac{t^n}{n}\rangle] = \exp(\sum_{n \geq 1} q_n \frac{t^n}{n}) ,$$

which is the e.g.f. for the cycle index partition polynomials of the symmetric groups, a.k.a. the Stirling partition polynomials of the first kind, OEIS A036039.

The natural generalization of the Stirling polynomials of the second kind is via the umbralization, for $q_0 = 1$,

$$e^{\langle e^{q.t}-1 \rangle} = \exp[ \; \sum_{n \ge 1} \; q_n \frac{t^n}{n!} \; ], $$

giving the complete Bell polynomials of the Faa di Bruno composition formula, a.k.a. the Stirling partition polynomials of the second kind, A036040.

Including an extra variable as in

$$e^{x\langle e^{q.t}-1 \rangle} = e^{Bell.(q_1,..,q.;x)t}$$

and

$$e^{-x\langle\ln(1-q.t)\rangle} = e^{Cyc.(q_1,..,q.;x)t}$$

allows better tracking of block sizes and combinatorics within each partition polynomial. (Note $q_n$ could be a commuting sequence of wedge products, curvature forms, or matrices, just as well as integers, giving basic formulas for the Chern and Pontryagin characteristic class polynomials.)

Now for a little umbral mojo,

$$ e^{St1.(St2.(x)) \; t} = e^{St2.(x) \; \ln(1+t)}=e^{x \; (\exp(\ln(1+t))-1)} = e^{xt} \; ,$$

so the two sets of binomial Sheffer polynomial sequences are an inverse pair under umbral composition; i.e.,

$$St1_n(St2.(x)) = x^n = St2_n(St1.(x)) \; .$$

This is reflected in their lower triangular coefficient matrices being a matrix inverse pair. The same umbral compositional inverse and matrix relationships hold for any Sheffer polynomial sequence. For a pair of inverse/reciprocal Appell Sheffer sequences, this is due to their moment e.g.f.s being multiplicative inverses rather than compositional inverse. The generic Sheffer sequence has a mix of these conditions, belonging to the semidirect product of the Appell and binomial Sheffer subgroups.

**Actions of diff op often become more transparent with umbral calculus and/or underlying combinatorics are revealed**:

With $a.$ treated as independent of $x$ and $D = d/dx$ for $f(x)= e^{b.x}$,

$$e^{a.D_x} \; f(x) = f(a.+x) = e^{b.(a.+x)}.$$

For the Bernoulli number and polynomials (or any Appell Sheffer sequence),

$B_n(x) = e^{b.D} \; x^n = (b.+x)^n$,

$D \; B_n(x) = D \; (b.+x)^n = n \; (b.+x)^{n-1} = n \; B_{n-1}(x)$

$e^{a.D} \; B_n(x) = B(x + a.) = (x+b.+a.)^n = (B.(x)+a.)^n = (B.(a.)+x)^n \; .$

Adopt the additional notational convenience $(:AB:)^n := A^nB^n$. Alternatively, we could use $q.^n = q_n = A^nB^n$, but this would veil aspects/intuitions inherited from the properties of the derivative.

From our previous feat of umbral mojo,

$$St1_n(xD) = n! \; \binom{xD}{n} = ST1_n(ST2.(:xD:) = \; :xD:^n \; = x^nD^n,$$

consistent with the actions

$xD \; [x^n] = n x^n$,

$(1 + a.)^{xD} \; [x^n] = (1 + a.)^{n} \; x^n$,

and

$(1 + a.)^{xD} \; f(x) = \sum_{n \geq 0} a_n \; \binom{xD}{n} \; f(x) = e^{a.:xD:} \; f(x) = f((1+a.)x).$

The operational definition of the Stirling polynomials of the second kind is

$$(xD)^n = (St2.(:xD:))^n = St2_n(:xD:).$$

From this the e.g.f. given above for the polynomials can easily be derived, and, consequently,

$$e^{txD} \; f(x) = e^{t \; St2.(:xD:)} \; f(x) = e^{(e^t-1):xD:} \; f(x)$$

$$ =f((e^t-1)x + x) = f(e^t x),$$

a dilation, consistent with $e^{txD} \; x^n = e^{tn} \; x^n$.

The Lah polynomials $L_n(x)$ (the normalized, unsigned Laguerre polynomials of order -1, related to partitions of sets into ordered sets or lists) have several useful Stokes-Rodrigues diff op reps,

$$ x \; :Dx:^n \; x^{-1} = x \; D^n x^n x^{-1} = x^{-n+1} (xDx)^n x^{-1} = x^{-n} (x^2D)^n = L_n(:xD:) ,$$

and the e.g.f.

$$e^{L.(x)t} = e^{x \frac{t}{1-t}},$$

so

$$ e^{tx^2D} \; f(x) = e^{t \; :xL.(:xD:):} \; f(x) = e^{:\frac{tx^2}{1-tx}D:} \; f(x)$$

$$= f(\frac{tx^2}{1-tx} + x) = f(\frac{x}{1-tx}), $$

a special linear fractional, or Moebius, transformation.

(The inverse ops to the translation, $e^{tD}$; dilation, $e^{txD}$; and special linear fractional, $e^{tx^2D}$, transformations of SL2 are given by negating $t$, which is equivalent to additive ($t+(-t)=0$), multiplicative ($e^t \cdot 1/e^t =1$), and compositional inversion ($\frac{x}{1-tx}|_{x \to\frac{x}{1+tx}}=x$), respectively. SL2 becomes associated to the underlying combinatorics of the binomial Sheffer polynomials $St2_n(x)$ and $L_n(x)$ via the umbral symbolic calculus.)

**Interpolation methods and results of integral transforms are naturally suggested and expressed in umbral notation**:

Laplace transform of e.g.f. to o.g.f.s with $q_0=1$:

$$\int_{0}^\infty \; e^{q.xt} \; e^{-t} dt = \int_{0}^\infty \; e^{-t(1-q.x)} dt = \frac{1}{1-q.x}$$

Mellin transform interpolation to continuous indices (related to Ramanujan's master formula, e.g., the Bernoulli polynomials are essentially discrete samples of the Hurwitz zeta function):

$$ q_{-s} = (q.)^{-s} := \int_{0}^\infty \; e^{-q.t} \; \frac{t^{s-1}}{(s-1)!} \; dt$$

Newton interpolation to continuous indices:

$$q_s = (q.)^s = (1-(1-q.))^s = \sum_{m=0}^\infty \; (-1)^m \; \binom{s}{m} \;\sum_{k=0}^m \; (-1)^k \; \binom{m}{k} \; q_k$$

$$= \int_{0}^\infty \; e^{-(1-(1-q.))t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt= \int_{0}^\infty \; e^{-t} \; e^{(1-q.)t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$

$$= \int_{0}^\infty \; e^{-q.t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$

via analytic continuation. If $q_n = D_x^n$, the action of the integral on $H(x) \frac{x^\alpha}{\alpha!}$, where $H(x)$ is the Heaviside step function, gives a classic fractional calculus in terms of an analytically-continued (AC) Mellin convolution (essentially that for the AC Euler beta function integral) for the interpretation of $D_x^{s}$ for real or complex $s$.

See Boole, Blissard, Sylvester, Cayley, Appell, (Jensen?), Heaviside, Steffensen, Sheffer, Pincherle, Bell, Riordan, Rota, Taylor, Roman, Ray, and Lenart for contributions to and influences on the development of the umbral symbolic calculus.

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