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A. K. Kwasniewski

Publications and source records attributed to A. K. Kwasniewski.

At least 37 records · Page 2Linked to original sources
arXiv2005

On umbral extensions of Stirling numbers and Dobinski-like formulas

Umbral extensions of the stirling numbers of the second kind are considered and the resulting dobinski-like various formulas including new ones are presented. These extensions naturally encompass the two well known q-extensions. The further consecutive umbral extensions q-stirling numbers are therefore realized here in a two-fold way. The fact that the umbral q-extended dobinski formula may also be interpreted as the average of powers of random variable with the q-poisson distribution singles out the q-extensions which appear to be a kind of singular point in the domain of umbral extensions as expressed by corresponding two observations. Other relevant possibilities are tackled with the paper`s closing down questions and suggestions with respect to other already existing extensions while a brief limited survey of these other type extensions is being delivered. There the newton interpolation formula and divided differences appear helpful and inevitable along with umbra symbolic language in describing properties of general exponential polynomials of touchard and their possible generalizations. Exponential structures or algebraically equivalent prefabs with their exponential formula appear to be also naturally relevant.

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arXiv2005

Prefab posets` Whitney numbers

We introduce a natural partial order in structurally natural finite subsets the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers` triangular array are then calculated and the explicit formula for them is provided. Next - in the second construction - we endow the set sums of prefabiants with such an another partial order that their their bell like numbers include fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial newton coefficients and fibonacci numbers onto the infinity of their relatives among which there are also the fibonacci triad sequences and binomial-like coefficients (incidence coefficients included).

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arXiv2005

Cobweb posets as noncommutative prefabs

A class of new type graded infinite posets with minimal element are considered. These so called cobweb posets introduced recently by the present author provide a wide range of new noncommutative prefab combinatorial schema with characteristic graded subposets as primes. The schema are defined here via relaxing commutativity and associativity requirements imposed on the composition of prefabs by the fathers of this fertile concept. The construction and the very first basic properties of cobweb prefabs are pointed out in what follows. An another single valued commutative amd associative composision is also considered.

math.CO↗
arXiv2005

Information on some recent applications of umbral extensions to discrete mathematics

At the first part of the paper we show how specific umbral extensions of the Stirling numbers of the second kind result in new type of Dobinski-like formulas. In the second part among others one recovers how and why Ward solution of uncountable family of extended difference calculus nonhomogeneous equations extends to Ward-Appell polynomials case . Illustrative specifications to q-calculus case and fibonomial calculus case are made explicit due to the usage of the so called upside down notation for objects of extended finite operator calculus .

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arXiv2004

Extended finite operator calculus as an example of algebraization of analysis

A wardian calculus of sequences started almost seventy years ago constitutes the general scheme for extensions of the classical umbral operator calculus considered by many afterwards . At the same time this calculus is an example of the algebraization of the analysis here restricted to the algebra of formal series. This is a review article based on the recent first author contributions. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by prominent contributors to the domain.

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arXiv2004

$ψ$-Appell polynomials` solutions of an umbral difference nonhomogeneous equation

One discovers why the solution of generalized umbral calculus difference nonhomogeneous equation in the form recently proposed by the author extends here now to generalized appellian delta operator and corresponding polynomials case almost automatically. The reason for that is just the proper framework of the generalized finite operator calculus recently being developed by the present author.

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arXiv2004

More on combinatorial interpretation of the fibonomial coefficients

Combinatorial interpretation of the fibonomial coefficients as a number of choices of specific finite subsets of an infinite partially ordered set of not binomial type is proposed. This partially ordered set is here defined via characteristic matrix of the corresponding partial order relation . Relevance of the proposal to more general unification treatment is indicated.

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arXiv2004

The logarithmic fibbinomial formula

Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.

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arXiv2004

Fibonacci q-gaussian sequences

The summation formula within pascalian triangle resulting in the fibonacci sequence is extended to the $q$-binomial coefficients $q$-gaussian triangles.

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arXiv2004

On $ψ$-basic bernoulli-wardian polynomials

The wardian solution of any $ψ$-difference linear nonhomogeneous equation is found in the framework of the generalized finite operator calculus . Specifications to $q$-calculus case and the new one fibonomial calculus case are made explicit.

math.CO↗
arXiv2004

Combinatorial interpretation of the recurrence relation for fibonomial coefficients

Combinatorial interpretation of the fibonomial coefficients recently proposed by the present author results here in combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided with quite an exhaustive context where apart from plane grid coordinate system used several figures illustrate the exposition of statements and the derivation of the recurrence itself.

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arXiv2004

First contact remarks on umbra difference calculus references streams

The reference links to the modern classical umbral calculus before that known as blissardian symbolic method are numerous. Rota founded and then extended finite operator calculus has links to references which are plenty numerous. The references via links to the difference q calculus umbra way treaded or without even referring to umbra are giant numerous. These reference links now result in counting in thousands the relevant papers . The purpose of the present attempt is to offer one of the keys to enter the world of those thousands of references.

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arXiv2004

$ψ$-Poisson, $q$-Cigler, $ψ$-Dobinski, $ψ$-Rota and $ψ$-coherent states

Cigler simple derivation of usual and extended Dobinski formula is recalled and it is noted that both may be interpreted as averages of powers of random variables with the corresponding usual or extended Poisson distributions. In parallel more general formulas of extended operator umbral calculi origin are revealed . The formulas encompass both earlier cases as very specific ones.

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arXiv2004

Higher Order Recurrences for Analytical Functions of Tchebysheff Type

One parameter subgroups of the group of hyperbolons of volume one when exploited accurately allow one to define and investigate higher order hyperbolic-trigonometric generalization of corresponding polynomials. In parallel functions of roots of polynomials of any degree are studied as possible extension of symmetric functions considered one hundred twenty six years ago in the inspiring famous lucasian paper.

math.GM↗
arXiv2004

The second part of on duality triads' paper

The duality triads were defined in the preceding paper.(ArXiv: math.GM/0402260 v 1 Feb. 2004). Notation, enumeration of formulas and references is therefore to be continued hereby. In this paper Fibonomial triangle and further Pascal-like triangles including q-Gaussian one are given explicit interpretation as discrete time dynamical systems as it is the case with all duality triads.

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