## Compositional Inversion and Appell Sequences

Compositional Inversion and Appell Sequences is a brief set of notes (pdf) on the relation between a Lagrange compositional inversion of a function in terms of the coefficients of the reciprocal of the function and a general set of Appell polynomials (special Scheffer sequences). The partition polynomials for the inversion form an Appell sequence w.r.t. the coefficient of the linear degree of the reciprocal and have coefficients related to a refinement of the Narayana numbers (A134264 / A001263) and  enumeration of Dyck (A125181) and Lukasiewicz paths, non-crossing partitions, and certain types of trees.

At Play in an Orchard: Catalan, Riordan Appels and Lagrange Pairs reiterates the above notes and extends them to show how the Catalan numbers (OEIS-A000108)  are related to other special number sequences, such as the Fibonacci (A000045), Motzkin sums / Riordan (A005043), and Fine (A000957), through simple compositions, and through simple interpolation to associated polynomials, such as the Fibonacci polynomials  (A030528)  and those related to the Dyck / Lukasiewicz lattice paths, non-crossing partitions, and trees of A091867. In addition, it provides a procedure for constructing or identifying families related to other parent sequences.

## Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra

Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra: A brief note (pdf) on some relations among these topics, including the Catalan and Fuss-Catalan numbers. References are provided linking the analysis with the distribution of eigenvalues of random matrices, the Wigner semicircle law, and moduli spaces for marked Riemann surfaces.

For some notes on the Burgers equation (generalized form),  see Lectures on Partial Differential Equations (Chapter 4) by R. Salmon.

## Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform

Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform:   (pdf, under construction. ) Relations between differential operators represented in the two basis sets $x^k (\frac{d}{dx})^k$ and $x^k (\frac{d}{dx})_{x=0}^k$ and the underlying binomial transforms of the associated coefficients of the pairs of series are sketched as well as the relations among the Newton-Gauss interpolation of these coefficients, the action of the differential operators, and an associated modified Mellin and inverse Mellin transform pair. The associated Laguerre polynomials  and the Bell / Touchard  / Exponential polynomials are examined in this light. Connections of the associated Laguerre polynomials to the Witt Lie algebra and modular forms are established.

## Goin’ with the Flow: Logarithm of the Derivative

Goin’ with the Flow: Logarithm of the Derivative Operator is a pdf set of notes under construction on the relations between the commutator of the logarithm of the derivative operator, the Pincherle derivative,  Lie operator derivatives, and the two umbrally inverse pair of Sheffer sequences the Bell polynomials and falling factorials. The action of the commutator / Lie op derivative is examined on fractional derivatives of the Riemann-Liouville type as represented by the Kummer confluent  hypergeometric functions (generalized Laguerre functions). Matrix reps are presented that illustrate the matrix rep of the Lie operator derivative as a conjugation of the matrix representing the usual derivative op by the Stirling number matrices (first and second kinds). Relations to the n-simplices are also given.

## The Vandermonde Matrix

On the Vandermonde Matrix: Some miscellaneous notes under construction on the Vandermonde matrix related to various MathOverflow questions and OEIS entries. The entry above, Goin’ with the Flow …, also contains some discussion of the Vandemonde matrix in the last section.

## Fractional Calculus, Interpolation, and Traveling Waves

Fractional Calculus, Interpolation, and Traveling Waves: A note (pdf file) on the fractional integro-derivative (FID) regarded as an interpolation of the integral derivatives of the function acted upon–to be more precise, a sinc function interpolation, as in the Whittaker-Shannon interpolation formula, of the integral derivatives, properly normalized. Interference of traveling waves is used to model the action of the FID.

A discussion of the history of the fractional / operational calculus can be found in The Theory of Linear Operators From the Standpoint of Differential Equations of Infinite Order by  Harold T. Davis (Principia Press, 1936). Also see F. Mainardi and G. Pagnini’s  notes on The Role of Salvatore Pincherle in the Development of Fractional Calculus.

For some connections between fractional calculus, the exponential / Bell / Touchard polynomials (i.e., the Stirling numbers of the second kind), umbral Euler integrals, and confluent hypergeometric functions see the exercises in my notes in my entry below A Generalized Dobinski Relation and the Confluent Hypergeometric Fcts. and also the following notes, MathOverflow, and MathStackExchange questions and answers:

1) Goin’ with the Flow: Logarithm of the Derivative Operator (see entry above)

Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras: A short set of notes sketching some relationships among infinitesimal generators represented as differential operators and infinite-dimensional matrices, the Pascal triangle / pyramid, $SL_2$ conformal transformations, the Witt and Virasoro algebras, the Hermite and generalized Laguerre polynomials of order $-1/2$, the Dedekind eta function, and combinatorics of the underlying integer matrices, inspired by Alexander Givental’s presentation of Virasoro operators in “Gromov-Witten invariants and quantization of quadratic Hamiltonians.” Some potential connections to knots, flows, dynamics, and Poincare-Maass series are also mentioned.