33 research outputs found
Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas
Motivated by the resemblance of a multivariate series identity and a finite
analogue of Euler's pentagonal number theorem, we study multiple extensions of
the latter formula. In a different direction we derive a common extension of
this multivariate series identity and two formulas of Lucas. Finally we give a
combinatorial proof of Lucas' formulas.Comment: 11 pages, to appear in Discrete Mathematics. See also
http://math.univ-lyon1.fr/~gu
Some congruences involving central q-binomial coefficients
Motivated by recent works of Sun and Tauraso, we prove some variations on the
Green-Krammer identity involving central q-binomial coefficients, such as where is
the Legendre symbol and is the th cyclotomic polynomial. As
consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q
&\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose
2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for , the
first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence
modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to
appear in Adv. Appl. Mat
On Fourier integral transforms for -Fibonacci and -Lucas polynomials
We study in detail two families of -Fibonacci polynomials and -Lucas
polynomials, which are defined by non-conventional three-term recurrences. They
were recently introduced by Cigler and have been then employed by Cigler and
Zeng to construct novel -extensions of classical Hermite polynomials. We
show that both of these -polynomial families exhibit simple transformation
properties with respect to the classical Fourier integral transform
Some aspects of Fibonacci polynomial congruences
This paper formulates a definition of Fibonacci polynomials which is
slightly different from the traditional definitions, but which is related to the
classical polynomials of Bernoulli, Euler and Hermite. Some related congruence
properties are developed and some unanswered questions are outlined.
Keywords: Congruences, recurrence relations, Fibonacci sequence, Lucas sequences,
umbral calculus