Formal group laws and binomial Sheffer sequences

Given a compositional inverse pair f(x) and f^{-1}(x), i.e.,

f(f^{-1}(x)) = x,

with f(x) = e^{a. x} with  a_0 = 0a_1 = 1, and (a.)^n = a_n,  construct the binomial Sheffer sequence p_n(t) with the exponential generating function

e^{x p.(t)} = e^{t f^{-1}(x)}.

Then the associated formal group law (FGL) may be expressed as

FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)] = e^{a.[f^{-1}(x)+f^{-1}(y)]}

= e^{a. D_t}|_{t=0} \;\; e^{t[f^{-1}(x)+f^{-1}(y)]}

= f(D_t)|_{t=0} \;\; e^{t[f^{-1}(x)+f^{-1}(y)]}

= f(D_t)|_{t=0} \;\; e^{t f^{-1}(x)} e^{tf^{-1}(y)}

= \sum_{j \geq 0 } \; \; \sum_{k \geq 0 } \; \; \frac{x^j}{j!} \frac{y^k}{k!} \; f(D_t)|_{t=0} \;\; p_j(t)p_k(t).

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An lnfinite Wronskian Matrix, Binomial Sheffer Polynomials, and the Lagrange Reversion Theorem

Form the infinite Wronskian matrix W(x,y) with elements

W_{j,k} = D_x^{j-1}\frac{[y \cdot h(x)]^k}{k!}.

A generating function for this matrix is

e^{\alpha D_x} e^{\beta y h(x)} = e^{\beta y h(\alpha+x)}= G

with k! \; M_{j,k} = D_\alpha^{j-1} \; D_\beta^k \; G \; |_{\alpha=\beta=0}.

If h(0) = 0 = h^{(-1)}(0), then also

G = e^{(\alpha+x)p.(\beta y)},

where (p.(y))^n = p_n(y) = \sum_{m=0}^n \; p_{n,m} \; y^m is a binomial Sheffer sequence of polynomials.

Then in this particular case,

\; W_{j,k} = \sum_{m \ge 0} \frac{x^{m-j+1}}{(m-j+1)!} \; p_{m,k} \; y^k

and so is the product of an upper triangular Toeplitz matrix of divided-powers in x, whose rows are the shifted summands of the Taylor series for e^x , and the Sheffer polynomial summand matrix in y. For example, these are the 4 by 4 submatrices:

\begin{bmatrix} 1 & x & x^2/2!& x^3/3! \\ 0 & 1 & x & x^2/2! \\ 0 & 0 & 1 & x\\ 0 & 0 & 0 & 1 \end{bmatrix}

\begin{bmatrix} p_{0,0} & 0 & 0 & 0 \\  p_{1,0} & y \;p_{1,1} & & 0 \\ p_{2,0} & y \; p_{2,1} & y^2 \; p_{2,2} & 0\\ p_{3,0} & y \; p_{3,1} & y^2 \; p_{3,2} & y^3 \; p_{3,3}\end{bmatrix} .

By inspection,

W_{j,k} = D_x^{j-1} \; y^k \; C_k(x)

where C_k(x)= \sum_{n. \ge 0} \; p_{n,k} \; x^n/n! = (h(x))^k/k!, the e.g.f. for the k-th column of the Sheffer matrix.

Revisiting the Lie infinigens of previous posts, we have, for u=h(x) and g(u)=1/(h^{(-1)}(u))^{'},

W_{j,k} = (g(u)D_u)^{j-1}\frac{[y \cdot u]^k}{k!} |_{u=h(x)},

and, consistently,

G = e^{\alpha g(u) D_u}\; e^{\beta y u} \; |_{u=h(x)} = e^{\beta y h(\alpha + h^{-1}(u))} \; |_{u=h(x)}.

The trace for the general matrix,

Tr[W] = \sum_{n \ge 0} W_{n,n} = \sum_{n \ge 0} \; D_x^{n-1}\frac{[y \cdot h(x)]^n}{n!}

with D^{-1} 1 = x , appears in several guises (see the earlier post The Lagrange Reversion Theorem and the Lagrange Inversion Formula), changing colors but maintaining the same basic form, in related but distinct formulations for compositional inversion and, therefore, pops up in the analyses of formal group laws; antipodes for Hopf algebras; combinatorics of forests of tree graphs; convex polytopes; moduli spaces of marked discs and punctured Riemann spheres; Feynman graphs for quantum fields; solutions of nonlinear PDEs, such as the inviscid Burgers’ equation; Hirzebruch genera; and umbral, or finite operator, calculus.

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Witt Differential Generator for Special Jack Symmetric Functions / Polynomials

Exploring some relations among the multinomial coefficients of OEIS A036038 and the compositional inversion formulas of A134264, A248120, and A248927, related to numerous combinatorial structures and areas of analysis, including noncrossing partitions and free probability,  I came across the Jack symmetric functions J_n^{\alpha}(x_1,x_2, ...) in an infinite number of variables as presented in “Some combinatorial properties of the Jack symmetric functions” by Stanley.

Equation a) on page 80 of his paper leads to an umbral generating formula for the related Jack symmetric polynomials (JSP)

 J_n^{\alpha}(x_1,x_2, ...,x_{n+1}) = (\hat{x}_1 + \hat{x}_2 + \cdots + \hat{x}_{n+1})^n,

where \hat{x}_k is to be regarded as a regular variable until the expression is reduced to monomials at which time it is to be evaluated as  \hat{x}_k^j = s_j (\alpha) \; x_k^j with s_j(\alpha)  =  1 (1+\alpha)(1+2 \alpha) \cdots (1+(j-1) \alpha), essentially the row polynomials of A094638 comprised of the Stirling numbers of the first kind.  For example,

J_2^{\alpha}(x_1,x_2,x_3) = (\hat{x}_1 + \hat{x}_2  + \hat{x}_3)^2

= \sum_{k=1}^3 \; \hat{x}_k^2 \; + \; 2 \; \sum_{i,j=1 ; i < j}^3 \; \hat{x}_i \; \hat{x}_j = m_{[2]}(\hat{x}_1,\hat{x}_2,\hat{x}_3) \; + \; 2 \; m_{[1,1]}(\hat{x}_1,\hat{x}_2,\hat{x}_3)

= s_2(\alpha) \; m_{[2]}(x_1,x_2,x_3) \; + \; 2 \; s_1(\alpha) \; s_1(\alpha) \; m_{[1,1]}(x_1,x_2,x_3)

= (1+\alpha) \; m_{[2]}(x_1,x_2,x_3) \; + \; 2 \; m_{[1,1]}(x_1,x_2,x_3) ,

where the polynomial has been expressed in the symmetric monomial polynomials (SMP), easily extended to an indefinite number of variables. The factors multplying the SMPs are the multinomial coefficients of A036038 and remain independent of the number of variables. Each summand of an SMP has the same configuration of exponents and subscripts, allowing the products of s_k (\alpha) to be easily determined and factored out after the umbral evaluation.

Similarly, umbral reduction of x_k^j = x_j transforms J_n^0(x_1,..,x_{n+1}) into the partitions of A248120.
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An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

(This is a duplicate of a Mathoverflow question posed in Oct 2014 that ran the gauntlet of the OCD cadre there–the demonstrative ones I assume avoid stepping on cracks in the sidewalk and become obstructive, hostile, and/or jealous at popular questions they can’t immediately circumscribe within their own rigid, limited views on how and what math should be done. See also this blog post and this MO-Q. In a couple of days the question got around 800 views, 9 upvotes, 7 downvotes, two or three close requests, several unconstructive/petty comments on rewriting/formatting, but no answers. Recognizing, from experience on MO, the futility of further refining the question and having pursued a plausible partial answer myself, I deleted most of it. Here is the full text so that I and others can easily follow related references.)

Prelude on Two Threads

Last month (Sept. 2014) the workshop New Geometric Structures in Scattering Amplitudes was hosted by CMI with the following partial overview:

Recently, remarkable mathematical structures have emerged in the study of scattering amplitudes, revealing deep links to algebraic geometry, arithmetic and combinatorics. There have been many exciting dual representations of amplitudes including

  •  polytopes, twistor diagrams and the positive grassmannian leading to the concept of the amplituhedron,
  •  polylogs, multizeta values and multiloop integrals,
  •  …

Each of these disparate ideas and methods have their own remarkable successes, and indeed have led to important progress in related areas of mathematics. They also face obstacles that they need to overcome in order to become important tools in resolving the most important open conjectures in the field and for the explicit construction of amplitudes.

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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)}.

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The Lagrange Reversion Theorem and the Lagrange Inversion Formula

From Wikipedia on the LRT, with

v(x,y) = x + y \; h(v(x,y)),

v(x,y) = x + \sum_{ n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x).

Letting x = 0 and w(y) = v (0,y),

w (y) = y \; h (w(y))  and h(y)=y/w^{(-1)}(y), giving

w(y) = \sum_{n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x) \; |_{x=0} \;,

the Lagrange inversion formula about the origin, whose expansion in the Taylor series coefficients of h(x) is discussed in OEIS A248927. See also A134685. For connections to free probability, free cumulants and moments, Appell sequences, noncrossing partitions, and other combinatorics, see A134264.

Let v^{(-1)}(x,y) be the inverse of v(x,y) w.r.t. to x . Then

x = v^{(-1)}(x,y) + y \; h(x), or

v^{(-1)}(x,y) = x - y \; h(x), and

v(x,y) = x + \sum_{ n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x) = x + \sum_{ n > 0} \frac {1}{n!} \; D_x^{n-1} [x-v^{(-1)}(x,y) ]^n.

The solution for the inverse of this last type is also presented in the post on the inviscid Burgers’ equation and the post  Generators, Inversion, and  Matrix, Binomial, and Integral Transforms.

The Laplace transform (LPT) argument in Appendix II of the Generators pdf can be briefly extended to derive the last form of the LRT above. (See below.)

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Dirac-Appell Sequences

The Pincherle derivative [T^n(L,R),R] = \frac {d}{dL}T^n(L,R)= n \cdot T^{n-1}(L,R)  is implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, comprised of the Dirac delta function and its derivatives, formed by taking the inverse Laplace transform of the polynomials of an Appell polynomial sequence.

The fundamental D-A sequence can be defined as the sequence \delta^{(n)}(x) with L = -x and R = D = d/dx and e.g.f. e^{tD} \delta(x) = \delta(x+t). Another example is provided by OEIS A099174 with the D-A sequence H_n(x) = h_n(D) \delta(x) where h_n(x) are the modified Hermite polynomials listed in the Example section of the entry. The modified Hermite polynomials can be characterized several ways:

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