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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