Results 1  10
of
11
Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K
 2008. EXPERIMENTATION AND COMPUTATION 19
, 2008
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elab ..."
Abstract

Cited by 41 (18 self)
 Add to MetaCart
If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing
On umbral extensions of Stirling numbers and Dobinskilike formulas
, 2008
"... Umbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the well known qextensions. The fact that the umbral qextended Dobinski formula may also be interpreted as the avera ..."
Abstract

Cited by 11 (9 self)
 Add to MetaCart
Umbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the well known qextensions. The fact that the umbral qextended Dobinski formula may also be interpreted as the average of powers of random variable Xq with the qPoisson distribution singles out the qextensions which appear to be a kind of bifurcation point in the domain of umbral extensions. The further consecutive umbral extensions of CarlitzGould qStirling numbers are therefore realized here in a twofold way.
Oneparameters groups and combinatorial physics
 Proceedings of the Symposium COPROMAPH3: Contemporary Problems in Mathematical Physics (PortoNovo
, 2003
"... In this communication, we consider the normal ordering of operators of the type ..."
Abstract

Cited by 11 (9 self)
 Add to MetaCart
In this communication, we consider the normal ordering of operators of the type
Suzuki T.: A new symmetric Expression of Weyl ordering
 Mod. Phys. Lett. A
"... For the creation operator a † and the annihilation operator a of a harmonic oscillator, we consider Weyl ordering expression of (a † a) n and obtain a new symmetric expression of Weyl ordering w.r.t. a † a ≡ N and aa † = N + 1 where N is the number operator. Moreover, we interpret intertwining form ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
For the creation operator a † and the annihilation operator a of a harmonic oscillator, we consider Weyl ordering expression of (a † a) n and obtain a new symmetric expression of Weyl ordering w.r.t. a † a ≡ N and aa † = N + 1 where N is the number operator. Moreover, we interpret intertwining formulas of various orderings in view of the difference theory. Then we find that the noncommutative parameter corresponds to the increment of the difference operator w.r.t. variable N. Therefore, quantum (noncommutative) calculations of harmonic oscillators are done by classical (commutative) ones of the number operator by using the difference theory. As a byproduct, nontrivial relations including the Stirling number of the first kind are also obtained. 1
On psiumbral extension of Stirling numbers and Dobinskilike formulas
 Advan. Stud. Contemp. Math
"... ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are therefore realized here in a twofold way. The fact that the umbral qextended Dobinski formula may also be interpreted as the average of powers of random variable Xq with the qPoisson distribution singles out the qextensions which appear to be a kind of ”bifurcation point ” in the domain of ψumbral extensions as expressed by Observations 2.1 and 2.2 as tackled with the paper closing down question.
Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials
"... We define a generalization of the Stirling numbers of the second kind, which depends on two parameters. The matrices of integers that result are exponential Riordan arrays. We explore links to orthogonal polynomials by studying the production matrices of these Riordan arrays. Generalized Bell number ..."
Abstract
 Add to MetaCart
We define a generalization of the Stirling numbers of the second kind, which depends on two parameters. The matrices of integers that result are exponential Riordan arrays. We explore links to orthogonal polynomials by studying the production matrices of these Riordan arrays. Generalized Bell numbers are also defined, again depending on two parameters, and we determine the Hankel transform of these numbers. 1
Ladder Operators and Endomorphisms in Combinatorial Physics
, 2009
"... Starting with the HeisenbergWeyl algebra, fundamental to quantum physics, we first show how the ordering of the noncommuting operators intrinsic to that algebra gives rise to generalizations of the classical Stirling Numbers of Combinatorics. These may be expressed in terms of infinite, but rowfi ..."
Abstract
 Add to MetaCart
Starting with the HeisenbergWeyl algebra, fundamental to quantum physics, we first show how the ordering of the noncommuting operators intrinsic to that algebra gives rise to generalizations of the classical Stirling Numbers of Combinatorics. These may be expressed in terms of infinite, but rowfinite, matrices, which may also be considered as endomorphisms of C[[x]]. This leads us to consider endomorphisms in more general spaces, and these in turn may be expressed in terms of generalizations of the ladderoperators familiar in physics.
Orderings of Operators in Quantum Physics
, 2003
"... For wellknown three orderings (normal ordering, antinormal ordering, and Weyl ordering) in Quantum Physics, we give a simple proof for the intertwining formulas among them. Moreover, for the creation operator a † and the annihilation operator a of a harmonic oscillator, we show Weyl ordering expre ..."
Abstract
 Add to MetaCart
For wellknown three orderings (normal ordering, antinormal ordering, and Weyl ordering) in Quantum Physics, we give a simple proof for the intertwining formulas among them. Moreover, for the creation operator a † and the annihilation operator a of a harmonic oscillator, we show Weyl ordering expression of (a † a) n as different types of polynomials with respect to the number operator N ≡ a † a. Nontrivial relations including the Stirling number of the first kind are also obtained. 1
Preface...................................
, 2005
"... Combinatorics of boson normal ordering and some applications ..."
Lucky 13th Exercises on Stirlinglike numbers and Dobinskilike formulas
, 803
"... Extensions of the Stirling numbers of the second kind and Dobinskilike formulas are proposed in a series of exercises for graduetes. Some of these new formulas recently discovered by me are to be found in A.K.Kwa´sniewski’s source paper [1]. These extensions naturally encompass the well known qext ..."
Abstract
 Add to MetaCart
Extensions of the Stirling numbers of the second kind and Dobinskilike formulas are proposed in a series of exercises for graduetes. Some of these new formulas recently discovered by me are to be found in A.K.Kwa´sniewski’s source paper [1]. These extensions naturally encompass the well known qextensions.The indicatory references are to point at a part of the vast domain of the foundations of computer science in ArXiv affiliation noted as CO.cs.DM.