Results 21 to 30 of about 142 (36)

A generalization of the binomial coefficients [PDF]

arXiv, 1995
We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici \`a d\'eterminer est la meilleure g\'en\'eralisation possible des factorielles et des coefficients du bin\^oome. On s'interesse
arxiv  

Applications of the classical umbral calculus [PDF]

Algebra Universalis 49 (2003), 397-434, 2001
We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.
arxiv  

On Simple Characterisations of Sheffer psi- polynomials and Related Propositions of the Calculus of Sequences [PDF]

Bull. Soc. Sci. Lett. Lodz Ser. Rech. Deform. 52, Ser. Rech. Deform. 36 (2002) pp.45-65, 2003
A calculus of sequences started in 1936 opened the way for future extensions of umbral calculus in its finite operator form. Because of historically established notation we call it the psi-calculus.It appears in parts to be almost automatic extension of the standard classical finite operator calculus.
arxiv  

Poisson, Dobinski, Rota and coherent states [PDF]

Bulletin de la Societe des Sciences et des Lettres de {\pounds}\'od\^e (54) Serie: Recherches sur les Deformations Vol. 45 (2004) 17-19, 2004
New q- Dobinski formula might also be interpreted as the average of specific q-powers of random variable X with the usual Poisson distribution.
arxiv  

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

Proc. Jangjeon Math. Soc. Vol. 7 (2), 2004 pp. 95-98, 2004
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
arxiv  

Cauchy type identities and corresponding fermatian matrices via non-comuting variables of extended finite operator calculus [PDF]

Proc. Jangjeon Math. Soc. Vol 8 (2005) no. 2. pp.191-196, 2004
New family of extended Cauchy type identities is found and related Fermat type matrices are provided ready for applications in extended scope. This is achieved due to the use specifically non-commuting variables of extended finite operator calculus introduced by the author few years ago.
arxiv  

Pascal like matrices - an accessible factory of one source identities and resulting applications [PDF]

Adv. Stud. Contemp. Math. 10 (2005) No. 2, pp. 111-120, 2004
The extension of pascalian like matrices depending on a variable from any field of zero characteristics are shown at work for the first time.
arxiv  

On $ψ$-basic bernoulli-wardian polynomials [PDF]

Bulletin de la Societe des Sciences et des Lettres de {\pounds}\'od\^e (54) Serie: Recherches sur les Deformations Vol. 45 (2004) 5-10, 2004
The wardian solution of any $\psi$-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.
arxiv  

Baxter Algebras and Umbral Calculus [PDF]

Adv. in Appl. Math. 27 (2001), 405-426, 2004
We apply recent constructions of free Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of the umbral calculus that include the classical umbral calculus in a family of $\lambda$-umbral calculi parameterized by ...
arxiv  

Baxter Algebras and Hopf Algebras [PDF]

Trans. Amer. Math. Soc., 355 (2003), 4639-4656, 2004
By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.
arxiv