The Heat Equation, the Quantum Harmonic Oscillator, the Hermite Polynomials, and Gaussian Integral Transforms

The Appell polynomial operator calculus of the Hermite polynomials provides a way to quickly derive, collate, and connect various properties of the Gaussian e^{-x^2} to important constructs in math and physics–the heat, or diffusion, equation; the quantum mechanical harmonic oscillator; the Fourier, the one-sided and two-sided Laplace, and the Mellin transforms of the Gaussian; and cumulants for the Gaussian probability density function for proof of the central limit theorem. This operator calculus revolves around the heat operator (differential form of the Segal-Bargmann transform), its inverse, and the Graves-Lie-Heisenberg-Weyl group/algebra based on the ladder operators–the raising and the lowering ops–of families of Hermite polynomials related via operator or functional conjugations.

pdf file: The heat equation, the quantum harmonic oscillator, the Hermite polynomials, and Gaussian integral transforms

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Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials

Classic relations among reps of a polynomial in terms of its zeros or a meromorphic function in terms of its zeros and poles, the symmetric polynomials/functions (the elementary, complete homogeneous, and power sum), matrix determinants, the Newton (Waring-Girard-Faber) identities, a couple of composition polynomial sequences (the Stirling partition polynomials of the first kind, a.k.a. the cycle index polynomials–CIPs– of the symmetric groups, and the Faber partition polynomials), and the formalism of Appell polynomial sequences are presented in the Word doc below and tested on the Riemann xi function as well as a combinatorial geometric interpretation of the CIPs based on walks on complete graphs, a.k.a. mystic roses.

Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials

(first draft, Word doc, large file)

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One of my favorite quotes is

“Go foward, faith will follow.” — d’ Alembert.

Every now and then you may need a pep talk based on some good examples of the importance of being earnest, brave, and persevering (with a good sense of humor to maintain perspective). Those asserting they are uniquely in possesion of some monolithic, revolutionary theory usually don’t need this, but us mortals with the right mix of healthy narcissism (hopefully based on accolades from others of experience), doubt (based on experience, again, of mistakes lurking in even the simplest calculations), and passion for understanding patterns/connections can benefit from such examples. Here’s one anecdote from an AMS bio on the first woman to win a Fields medal Maryam Mirzakhani (1977–2017):

Maryam’s perseverance stands out in our minds. In
the early spring of her last year in graduate school, she
rushed into one of our offices close to tears. She was
holding what must have been the twentieth draft of her
thesis, covered in McMullen’s famous red ink. “I am not
sure I will ever graduate,” she moaned, and she joked
that the last ten pages did not have as much red only
because McMullen’s pen must have run out of ink. After
commiserating for half an hour, Maryam steeled herself
and with her characteristic determination marched back
to her office for yet another round of editing.

Some other vignettes:

“A Visit to Hungarian Mathematics” by the late Reuben Hersh

Any bio on Hooke, Faraday, Fourier, Abel, Galois, Riemann, Green, Ramanujan, Grothendieck–all achieved much (four in short lives) despite deprivations–or on any number of Chinese, Russian, eastern European mathematicians/scientists oppressed by autocratic regimes and/or socioeconomic, gender, or other bias (including, of course, Marie Curie).

And d’ Alembert.

Yet the light penetrates the darkness.

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Newton’s Hyperbolic Appells

Newton, Laguerre, de Gua, Jensen, Turan, and Polya all derived certain inequalities involving the derivatives or coefficients of polynomials that must be satisfied for the polynomials to be hyperbolic, i.e., to have only real zeros. I reproduce Stanley’s proof of the basic Turan inequalities in more detail, point out some ramifications and other proofs of related inequalities, discuss the geometric import, and then give a toy example to explore to illustrate a set of hyperbolic polynomials and their kissing curve limits–non-hyperbolic bounding polynomials.

Newton’s Hyperbolic Appells (pdf)

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Appell Matrices

A summary of observations recorded in diverse sites over the years (stemming from OEIS A133314) on the general matrix algebra of the coefficient matrices of Appell polynomial sequences and the connections of that algebra to umbral compositional inversion, binomial convolution, and manipulation of exponential generating functions.

First draft (pdf):

Appell Matrices

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A Teaser on the Dobinski Formula and the Bell Polynomials, or Stirling Polynomials of the Second Kind


The Stirling numbers of the second kind were a first introduction to me of classic combinatorics in operational calculus. Later I became acquainted with their connection to Dobinski’s formula which can be generalized to obtain many different op calc/combinatorial identities. This is only a very brief teaser on the topics, on which I have posted several times here, in the OEIS, and a couple or more at MathOverflow.

With the Euler op x\frac{d}{dx} = xD (a.k.a. the state number operator in QM) and the convenient notation :xD:^n = x^nD^n, the Bell polynomials can be defined operationally as

(xD)^n = Bell_n(:xD:) =\sum_{k=0}^n \; Bell_{n.k} \; x^n \; D^n.

With (a.)^n=a_n=f(n)=f(x)|_{x=n}, the generalized Dobinski relation is a consequence of

f(Bell.(:xD:)) \; x^n=f(xD) \; x^n=f(n) \; x^n=a_n \; x^n=(a. \; x)^n \; ,


f(Bell.(:xD:)) \; e^x=e^x \; f(Bell.(x))=f(xD) \; e^x=e^{a.x}.

Transposing the exponential and noting e^{-x} \; e^{a.x} = e^{-(1-a.)x}, we have the generalized Dobinski formula

f(Bell.(x)) = e^{-x}\sum_{n\geq 0}f(n) \frac{x^n}{n!} = \sum_{n\geq 0}(-1)^n \bigtriangledown_{k=0}^n f(k) \frac{x^n}{n!}

with the finite difference

\bigtriangledown_{k=0}^n a_k = \sum_{k=0}^n (-1)^k \binom{n}{k} a_k = (1-a.)^n \; .

With f(x) = e^{t \; x}, we directly obtain the e.g.f. for the Bell polynomials (OEIS A048993, another reduced version is A008277)

e^{t \; Bell.(x)} = e^{x(e^t-1)}.

With f(x) = x^m,

(Bell.(x))^m = Bell_m(x) = e^{-x}\sum_{n\geq 0}n^m \frac{x^n}{n!} = \sum_{n\geq 0}(-1)^n \; \bigtriangledown_{k=0}^n k^m \; \frac{x^n}{n!} .


Bell_{m,n} = (-1)^n \bigtriangledown_{k=0}^n \; k^m \; .

Note the finite difference \bigtriangledown_{k=0}^n \; p_m(k) vanishes for any k>m for any polynomial p_m(x) of degree m just as for the analogous derivative. This follows by taking the derivative of both sides of the Dobinski formula with f(x) = p_m(x) since p_m(Bell.(x)) is also a polynomial of degree m.

An operational proof of the Dobinski formula using an inverse Mellin transform allows more general extensions. Rota gave a combinatorial interpretation of the basic Dobinski formula in “The Number of Partitions of a Set” and Pitman, a probabilistic interpretation in “Some Probabilistic Aspects of Set Partitions.”

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Reciprocity and Umbral Witchcraft: An Eve with Stirling, Bernoulli, Archimedes, Euler, Laguerre, and Worpitzky

Motivated by the appearance of the Eulerian polynomials in algebraic geometry, geometric combinatorics, and in some derivations of the Baker-Campbell-Hausdorff-Dynkin (BCHD) expansion, identities are generated using umbral Sheffer calculus couplings of the iconic inverse pair e^t-1 and \ln(1+x), naturally relating the Stirling polynomials of the first and second kinds; the Eulerian, the Bernoulli, and the generalized Laguerre polynomial sequences; the canonical and umbrally morphed Worpitzky identities (WI); and the combinatorics of the classic simple convex polytopes the permutahedra/permutohedra. Relatively new polynomial sequences are procreated as well and a not oft-noted genealogy drawn encompassing the Kummer confluent hypergeometric polynomials, the Swiss-knife polynomials, the Worpitzky triangles, and the f- and h-polynomials of the stellahedra (whose h-vectors enumerate the positroids, or non-negative Grassmannians). These polynomials, their e.g.f.s (or related o.g.f.s), their umbral generalizations and differential ops and identities, or similar, are often found in presentations on symmetric functions, quantum field theory, combinatorial Hopf algebras, L-function theory, characteristic classes, volumes of polytopes, and the Magnus expansion and other methods of Lie/geometric numerical integration.

Continue reading

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Differintegral Ops and the Bernoulli and Reciprocal Polynomials

A short pdf on differintegral operators that generate the Bernoulli polynomials and their elegant consorts the Reciprocal polynomials, which form an inverse pair under umbral composition (mostly reprising notes in earlier posts):

Differintegral Ops and the Bernoulli and Reciprocal Polynomials

Related stuff (in addition to the compilation in the pdf):

Notes on the theorem of Baker-Campbell-Hausdorff-Dynkin” by Michael Mueger

The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations” by Strichartz

A note on the Baker–Campbell–Hausdorff series in terms of right-nested commutators” by Arnal, Casas, and Chiralt

Interaction between Lie theory and algebraic geometry” by Shilin Yu

“Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra” by Matthias Beck and Sinai Robins

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Juggling Zeros in the Matrix (Example II)

This is a sequel to my last post Skipping over Dimensions, Juggling Zeros in the Matrix with a second example: Laguerre polynomials of order  -1/2,  OEIS A176230.

Juggling Zeros in the Matrix (Example II)


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Skipping over Dimensions, Juggling Zeros in the Matrix

Slipping between dimensions can often simplify a problem. A fundamental example is side-stepping off the real line to the complex plane to find the zeros of a polynomial. In the cases I have in mind, this jump amounts to aerating a matrix, i.e., adding interleaving zeros, and recognizing the matrix could be the alternating rows/polynomials of an Appell matrix made up of the coefficients of an Appell sequence of polynomials, a matrix that can be represented by diagonal multiplication of the Pascal matrix by a column vector with alternating zeros. This allows us to easily find an analytic expression for the matrix inverse and to reveal algebraic and differential relations among the columns and rows of the matrix that are not transparent otherwise.

Example matrices presented in the following pdf are related to Appell polynomials initially explored by Jensen and, subsequently, Polya in their explorations of the real zeros of (more accurately, the hyperbolicity of) Fourier transforms related to the Landau-Riemann xi function, and to the Appell Hermite polynomials and associated Laguerre polynomials found in quantum mechanics and Heisenberg algebras.

Skipping over Dimensions, Juggling Zeros in the Matrix

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