Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering

A set of identities that encapsulates relations among the Bernoulli numbers, the Stirling numbers of the first and second kinds, and operators related to the umbral calculus of Blissard and his contemporaries:



=\frac{1}{<\; e^{\hat{B}.(0)\frac{d}{d(xD)}}\;>}=\frac{1}{< \; e^{\hat{B}.(0)\;nad_{\ln(D)}}\; >}=\; <\;\frac{1}{1+\hat{B}.{(0)}\frac{d}{d(:xD:)}}\;>.


The symbols with periods as subscripts are to be treated as umbral variables to be evaluated umbrally, i.e.,  \; (a.)^n=a_n \;, at the level indicated by the \; < \cdots >\;, that is, within these delimiters, the umbral variables may be treated as regular commuting variables that simply multiply together. Once a series in these variables is established within the delimiters with each summand containing only a single power of the umbral variable, the single power can be lowered to the subscript position and the umbra assigned whatever value (numerical, polynomial, operator, whatever) consistent to that subscript value throughout the series.

D= \frac{d}{dx}\;, the derivative, by definition \;(:xD:)^n=x^nD^n\;, \; \ln(x) \; is the natural logarithm, and


This last set of identities follows from arguments in the notes “Goin’ with the Flow” below, the binomial translation property of the Bernoulli polynomials, (B.(x)+y)^n=\sum_{k=0}^{n} \binom{n}{k} B_k(x)\;y^{n-k}=B_n(x+y) , and their defining property connecting them to the tangent space for formal power series f(B.(x+1))-f(B.(x))=D \; f(x).

The Bernoulli numbers B_n(0) together with the binomial translation property also uniquely define the Bernoulli polynomials, and are related to the Riemann zeta by the Mellin transform of their e.g.f. (with a sign introduced), while the Mellin transform of the  e.g.f. (mod sign) of the Bernoulli polynomials gives the Hurwitz zeta function. Some useful identities which follow from the Mellin transform relations are (-1)^n \frac{B_{n+1}(0)}{n+1}=-\frac{B_{n+1}(1)}{n+1}=\zeta(-n), which can be related to \zeta(2n) with the functional, or reflection, formula of the Riemann zeta, except for \zeta(0)=B_1(0)=-1/2, which is the odd man out.

The umbral compositional inverses  of the Bernoulli polynomials  are defined by B_n(\hat{B}.(x))=x^n and vice versa, and this, together with the Appell property, means that their e.g.f.s are multiplicative reciprocals as evident in the identities above. Specifically, the reciprocal polynomials are \hat{B}_n(x)=\frac{(1+x)^{n+1}-x^{n+1}}{n+1}.

(An aside.) Using differential operators in the rich tradition discussed by Harold Davis in his book The Theory of Linear Operators, and with a grasp of the fundamentals of the generalized shift op and Gauss-Newton series or interpolation, you can operate on both sides of the iconic Euler integral for the gamma function, using not the values of the Bernoulli numbers but only the fact that the polynomials are comp inverses of the reciprocal polynomials, to obtain the Mellin transform for the Hurwitz zeta function, or operate on \ln(s+1)! expressed as the log of the Weierstrass expression for the gamma function to get the digamma function as a power series in the values of the Riemann zeta at n \ge 2. A second differential op that is equivalent to the operational definition of the Bernoulli polynomials acting on functions analytic at \; x=1 \; is

D_x=\exp[-(1-B.(x+1))D_y] -exp[-(1-B.(x))D_y] \;|_{y=1} \;.

 Try acting on x^{-s} \; with it using the generalized shift operation e^{a. \; D}\; f(x)= f(a.+x) \; and the generalized binomial expansion to obtain the Hurwitz zeta function in its usual series rep, as a Mellin transform, and as a Gauss-Newton interpolation series of the Bernoulli polynomials. For operation on functions analytic at \; x=0 \;, e.g., \; x^n\;, this consistently gives the same result as

D_x = \exp[B.(x+1) \; D_y] - \exp[B.(x) \; D_y] \; |_{y=0} \;.

You can check that both formulations are equivalent for s=-n \; by noting the binomial transform (with signs) is an involution, i.e., self-reciprocal, i.e., (1-(1-a.))^n=a_n \; and that t^{s-1}/(s-1)! \; in the Mellin transform can be replaced by the derivatives of the Dirac delta funtion for such values of s.

The end result is that you can see precisely the true nature of the Mellin transform in interpolating the coefficients of e.g.f.s (or as generating a reciprocal s-space with a curve whose trajectory intersects with the coefficients at integer values of s)  and are now justified in regarding the Hurwitz zeta function as an interpolation and therefore generalization of the Bernoulli polynomials, and you will be in the company of Euler, Riemann, and Ramanujan with his Master Theorem / Formula (not such bad company). The machinations involved in regularizing the Mellin transform to analytically  continue it are in accord with displaying this curve. Think of the Euler integral for the gamma function. You subtract out the lower coefficients, or powers, that lead to singularities in the integrand of the Mellin transform to view the nature of the curve for values of Re(s) < 0, as the constant one. Same for the Bernoulli polynomials a.k.a. the Hurwitz zeta. To obtain the Riemann zeta, simply assign x=1.

(Back to the main points.)  This is all closely related to the two sets of Stirling numbers: the Bell polynomials, (xD)^n=\phi_n(:xD:)\;, whose coefficients are the Stirling numbers of the second kind, and their umbral compositional inverses, the falling factorials, whose coefficients are the Stirling numbers of the first kind, \; (\phi.(:xD:))_n= ((xD).)^n=(xD)_n=\frac{(xD)!}{(xD-n)!}=( :xD:)^n= x^nD^n \;, so also \;\phi_n((xD).)=\phi_n(:xD:)=(xD)^n. These can be taken as the operational definitions of these binomial Sheffer sequences.

You should also be careful with taking the powers of the expressions in the top identities. The eval for the Bernoulli umbral variables must either be done first or the variables treated as in my entry on the Todd class and the Hirzebruch criterion. The paper by B. Taylor that I mention in that entry also has an informative discussion of this in terms of moments of independent random variables.

The fact that the Bell polynomials and the falling factorials are an umbral compositional inverse pair is reflected in the two lower triangular matrices of those numbers being an inverse pair as well as the two functions that define the pair of binomial Sheffer polynomial sequences. The functions h(t)= e^t-1 and h^{(-1)}(t)=\ln(1+t) give e^{x(e^t-1)}=e^{t \phi.(x)},  and e^{x\; \ln(1+t)}=(1+t)^x=e^{t\;(x).} \; .

This also leads to  the operator  compositional inverse pair \; \frac{d}{d(:xD:)}=e^{\frac{d}{d(xD)}}-1\;, the lowering op for \;(xD)_n=(:xD:)^n=x^nD^n\;,  and \;\frac{d}{d(xD)}=\ln(1+ \frac{d}{d(:xD:)})\;, the lowering op for (xD)^n=\phi_n(:xD:).

The formal group law FGL associated with the Stirling compositional inverse pair is the multiplicative FGL

FGL(y,z)=h[h^{(-1)}(y)+h^{(-1)}(z)]=y+z+y \; z,

which in op form becomes


All these Lie derivatives map to matrix relations, as discussed in “Goin’ with the Flow” and entries on the OEIS, with the infinitesimal Lie generator of the Pascal matrix and the Pascal matrix itself playing central roles along with conjugation (guage transformation) by the two fundamental, mutually orthogonal Stirling matrices.  Essentially, \frac{d}{d(:xD:)} is mapped to dP, the infinitesimal matrix (nilpotent to the order of its rank if truncated, with vanishing trace and determinant) of the Pascal matrix. However, these matrices are conjugated with the mutually orthogonal pair of Stirling matrices. These results lead also to the matrix OEIS A238363 with its associations to the Coxeter A_n group and Bernoullis through A074909, A135278, and A130534.  The Bernoulli polynomials have the e.g.f. \frac{t}{e^t-1}e^{xt}=e^{tB.(x)} \; , the reciprocal polynomials, \frac{e^t-1}{t}e^{xt}=e^{t \hat{B}.(x)} \;, associated with A074909, and a third e.g.f. in the top identities is  \frac{ln(1+t)}{t}e^{xt}=e^{tM.(x)} \; , which is an Appell e.g.f. associated to A238363. Taking powers of these matrices is relatvely straightforward since they involve conjugation of an infinitesimal generator.

The operator relations hold just as well for any lowering op L and raising op R acting on its associated sequence, p_n(x) with p_0(x)=1 \;, L\; p_n(x)= n \; p_{n-1}(x) \;, and R \; p_n(x) = p_{n+1}(x) \;. Simply replace x\; with R \; and D \; with L \;  everywhere. The last section of Goin’ with the Flow presents raising and lowering ops related to geometric operations on the simplices. The exponentiated raising op then gives bivariate polynomials for the face polynomials of the simplices, essentially the polynomials of A074909, A135278, and A130534.

These relations run parallel to the classic relations for the finite difference operator:

\displaystyle \triangle f(x) = f(x+1) - f(x) = (e^D-1) \; f(x) \; \; and

\displaystyle \frac{D}{\triangle} = \frac{\ln(1+ \triangle)}{\triangle} = \frac{D}{e^D-1} .

Related stuff:

“Hopf algebras in dynamical systems” by Carinena, Ebrahimi-Fard, Figueroa, and Gracia-Bondi

“Central differences, Euler numbers and symbolic methods” by Dowker

Poincare and Lie groups” by Schmid

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