On Fourier integral transforms for qq-Fibonacci and qq-Lucas polynomials

We study in detail two families of $q$-Fibonacci polynomials and $q$-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 $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform

Lucas' theorem: its generalizations, extensions and applications (1878--2014)

In 1878 \'E. Lucas proved a remarkable result which provides a simple way to compute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms of the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the $p$-adic expansions of nonnegative integers $n$ and $m$, then \begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}. \end{equation*}} The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of Lucas}), plays an important role in Number Theory and Combinatorics. In this article, consisting of six sections, we provide a historical survey of Lucas type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas like theorems for some generalized binomial coefficients, and some their applications. In Section 1 we present the fundamental congruences modulo a prime including the famous Lucas' theorem. In Section 2 we mention several known proofs and some consequences of Lucas' theorem. In Section 3 we present a number of extensions and variations of Lucas' theorem modulo prime powers. In Section 4 we consider the notions of the Lucas property and the double Lucas property, where we also present numerous integer sequences satisfying one of these properties or a certain Lucas type congruence. In Section 5 we collect several known Lucas type congruences for some generalized binomial coefficients. In particular, this concerns the Fibonomial coefficients, the Lucas $u$-nomial coefficients, the Gaussian $q$-nomial coefficients and their generalizations. Finally, some applications of Lucas' theorem in Number Theory and Combinatorics are given in Section 6.Comment: 51 pages; survey article on Lucas type congruences closely related to Lucas' theore

Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$1+\frac 12 +\frac 13+...+\frac{1}{p-1}$$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2}$$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely, we present and compare several generalizations and extensions of Wolstenholme's theorem obtained in the last hundred and fifty years. In particular, we present more than 70 variations and generalizations of this theorem including congruences for Wolstenholme primes. These congruences are discussed here by 33 remarks. The Bibliography of this article contains 106 references consisting of 13 textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of Integer Sequences. In this article, some results of these references are cited as generalizations of certain Wolstenholme's type congruences, but without the expositions of related congruences. The total number of citations given here is 189.Comment: 31 pages. We provide a historical survey of Wolstenholme's type congruences (1862-2012) including more than 70 related results and 106 references. This is in fact version 2 of the paper extended with congruences (12) and (13