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.

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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.

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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)

2) Pochhammer symbol of a differential, and hypergeometric polynomials

3) Lie group heuristics for a raising operator for (-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}

4) Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional caluclus

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Infinigens, the Pascal Triangle, and the Witt and Virasoro Algebras

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.

(Revised Dec. 11. 2013.)

Added 2/2014: For some discussion of the relations between the formal group laws and local Lie groups introduced in this paper, see Applications of Lie Groups to Differential Equations (2nd Ed., pg. 18) by Peter Olver.

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Depressed Equations and Generalized Catalan Numbers

Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers is a set of notes on the the relation of generating functions of the generalized Catalan numbers, e.g., OEIS-A001764, to the compositional inverse of G(x,t)= x + t \: x^n and the tangent envelope of associated discriminant curves.

Added 2/2014: See also “Polygonal Dissections and Reversions of Series” by Alison Schuetz and Gwyn Whieldon for relations between the generalized Catalan (Fuss-Catalan) numbers and dissections of polygons.

The original title was “Discriminating Deltas, Depressed Equations, and Fussy Catalan Numbers,” but some Russian (Vlad the Impaler) with an inferiority-superiority complex took umbrage at this play of words at the expense of a fellow comrade and made a fuss about it.

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Riemann’s Jump Function J(x) for the Primes

Dirac’s Delta Function and Riemann’s Jump Function J(x) for the Primes presents Riemann’s jump function for counting the primes as introduced in H. M. Edward’s Riemann’s Zeta Function (Dover, 2001), couched in terms of the Dirac delta function and the inverse Mellin transform.

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A Generalized Dobinski Relation and the Confluent Hypergeometric Fcts.

The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions   presents a generalized Dobinski relation umbrally incorporating the Bell / Touchard / Exponential polynomials that is defined operationally through the action of the operator  f(x d/dx) on a modified inverse Mellin transform. Relations to the Dirac delta function/operator and, through an appropriate choice of f, the confluent hypergeometric functions, one set of which are the generalized Laguerre functions, are sketched and finally some exercises presented.

The exercises include formulas for the Riemann-Liouville and Weyl fractional integroderivatives (differintegrals) and their relations to an umbral Euler integral for the gamma function and the Kummer and Tricomi confluent hypergeometric functions.

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