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2022 vasario 18 d., 21:56 atliko AndriusKulikauskas -

Ištrintos 0-69 eilutės:

The Meixner polynomials involve a substitution {$x\to px$} where {$p=\sqrt{d^2-4g}$} so that the links are balanced by the kinks. How does that relate to Poincare invariance?

Hermite and Kravchuk
* How are Hermite and Kravchuk polynomials related as limiting cases?
* In what sense are Kravchuk polynomials a special case of Meixner polynomials?
* What would {$\alpha$} and {$\beta$} be for Meixner polynomials?
* Why exactly are the Hermite polynomials not orthogonal with regard to the binomial distribution?
* Why do Meixner, Charlier, Kravchouk polynomials have discrete probability mass functions and Hermite, Laguerre, Meixner-Pollaczek have continuous probability density distributions?
* [[https://www.fis.unam.mx/~bwolf/Articles/162.pdf | Mesuma K. Atakishiyeva, Natig M. Atakishiyev, Kurt Bernardo Wolf. Kravchuk polynomials and irreducible representations of the rotation group SO(3)]] related to quantum harmonic oscillator and to SU(2). Mentions that Charlier and Meixner are related to SU(1,1).
* [[https://spark.siue.edu/cgi/viewcontent.cgi?article=1008&context=siue_fac | G.Staples. Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras]]
** Kravchuk polynomials can be interpreted as matrix elements for representations of SU(2). T. Koornwinder, Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal., 13 (1982), 1011–1023.
** In quantum theory, Kravchuk matrices interpreted as operators give rise to two new interpretations in the context of both classical and quantum random walks. The latter interpretation underlies the basis of quantum computing. P. Feinsilver, J. Kocik, Krawtchouk matrices from classical and quantum random walks, Contemporary Mathematics, 287 (2001), 83–96.
** In the context of the classical symmetric random walk, Kravchuk polynomials are elementary symmetric functions in variables taking values ±1. P. Feinsilver, R. Schott, Krawtchouk polynomials and finite probability theory, In: H. Heyer (ed.) Probability Measures on Groups X (Olberwolfach 1990), pp. 129–135, Plenum, New York, 1991.
* [[https://www.researchgate.net/publication/232923344_Recurrences_and_explicit_formulae_for_the_expansion_and_connection_coefficients_in_series_of_classical_discrete_orthogonal_polynomials | Doha, Ahmed. Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials]] Has recurrence formulas for Kravchuk, Meixner, Charlier polynomials.

[[https://en.wikipedia.org/wiki/Random_matrix#Gaussian_ensembles | Gaussian ensembles]], in random matrix theory, have orthogonal, unitary, symplectic versions. They involve symmetric, Hermitian, Hermitian quaternionic matrices. They are said to relate to Hermite polynomials, but how?

How can I write the spherical harmonics, usually written in terms of {$r, \Phi, \theta$}, in terms of {$r, x, y, z$} ? And in what sense are they orthogonal polynomials?

Interpreting how an observable as an operator that is acting as a shift upon the Hermite polynomial (the eigenfunction)
* Potential - is a term that is a second derivative of space-time
* Observable (energy) is (derivative of space-time)(derivative of information)
* Second derivative of information gives zero (after imposing orthogonality)

There is slack, gap, in defining orthogonal polynomials, bases for linear subspaces, and so on. What is the relationship between that freedom, the cracks in the top down approach, and canonical choices?

[[https://reader.elsevier.com/reader/sd/pii/S0195669800904591?token=CC946FD202A0E8BA9930B094A851F59D6588F203478939EAA902903628531692569AEF3A2E86B7B82A32A399F4E8337D&originRegion=eu-west-1&originCreation=20220215192340 | Kim, Zeng. A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials.]]

[[https://www-users.cse.umn.edu/~stant001/publist.html | Dennis Stanton. Publication list.]]
* [[https://www-users.cse.umn.edu/~stant001/PAPERS/op.pdf | Stanton. Orthogonal polynomials and combinatorics.]]
* [[https://www-users.cse.umn.edu/~stant001/PAPERS/corteelkimstantonJuly10.pdf | Corteel, Kim, Stanton. Moments of Orthogonal Polynomials and Combinatorics.]]

Three-dimensional space
* One distinguishable dimension (in which motion occurs)
* Two indistinguishable dimensions (bosonic?)

Feynman diagrams
* https://indico.cern.ch/event/421552/sessions/170249/attachments/884224/1242865/feynman.pdf
* https://www.pas.rochester.edu/assets/pdf/undergraduate/Calculating_Transition_Amplitudes_from_Feynman_Diagrams.pdf

[[https://www.ams.org/journals/tran/1957-085-02/S0002-9947-1957-0091566-1/S0002-9947-1957-0091566-1.pdf | Karlin, McGregor. The Differential Equations of Birth-and-Death Processes, and the Stieltjes Moment Problem]]

[[https://www.ams.org/journals/tran/1957-086-02/S0002-9947-1957-0094854-8/S0002-9947-1957-0094854-8.pdf | Karlin, McGregor. The Classification of Birth and Death Processes]]

[[https://msp.org/pjm/1958/8-1/pjm-v8-n1-p08-p.pdf | Karlin, McGregor. Many server queueing processes with Poisson input and exponential service time.]]

[[https://en.wikipedia.org/wiki/M/M/1_queue | M/M/1 Queue]]

We can write a probability mass function dependent on ''t'' to describe the probability that the M/M/1 queue is in a particular state at a given time. We assume that the queue is initially in state ''i'' and write ''p''''k''(''t'') for the probability of being in state ''k'' at time ''t''. Then<ref>{{cite book | title = Queueing Systems Volume 1: Theory | first1=Leonard | last1=Kleinrock | author-link = Leonard Kleinrock | isbn = 0471491101 | year=1975 | page=77}}</ref>

{$p_k(t)=e^{-(\lambda+\mu)t} \left[ \rho^{\frac{k-i}{2}} I_{k-i}(at) + \rho^{\frac{k-i-1}{2}} I_{k+i+1}(at) + (1-\rho) \rho^{k} \sum_{j=k+i+2}^{\infty} \rho^{-j/2}I_j(at) \right]$}

where {$i$} is the initial number of customers in the station at time {$t=0$}, {$\rho=\lambda/\mu$}, {$a=2\sqrt{\lambda\mu}$} and {$I_k$} is the modified Bessel function of the first kind. Moments for the transient solution can be expressed as the sum of two monotone functions.

<ref>{{Cite journal | last1 = Abate | first1 = J. | last2 = Whitt | first2 = W. | author-link2 = Ward Whitt| doi = 10.1007/BF01182933 | title = Transient behavior of the M/M/l queue: Starting at the origin | journal = [[Queueing Systems]]| volume = 2 | pages = 41–65 | year = 1987 | url = http://www.columbia.edu/~ww2040/TransientMM1questa.pdf}}</ref>

Viennot video
* Combinatorial intermpretation of Favard theorem (about the recurrence formula for orthogonal polynomials). The duality between the combinatorics of moments and the combinatorics of the coefficients of orthogonal polynomials is the duality between elementary symmetric functions and homogeneous symmetric functions. (But which way?)
* Tridiagonal matrices are related to PASEP, random matrix theory, orthogonal polynomials.

Fivesome
* The conceptual foundation from which we can derive Minkowski space, special relativity, quantum field theory.

How could general relativity arise from the global quantum as grounded in the foursome?

Relate Viennot's histoire with automata theory.

Interpret Viennot's tableaux by rotating them 45 degrees so that they are like the Young tableaux and the binomial triangle.